Science in the Looking Glass: What Do Scientists Really Know?

Science in the Looking Glass

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Science in the Looking Glass What Do Scientists Really Know?

E. Brian Davies Department of Mathematics King’s College London

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Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York © Oxford University Press, 2003 The moral rights of the authors have been asserted Database right Oxford University Press (maker) First published 2003 First published in paperback 2007 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available Typeset by Newgen Imaging Systems (P) Ltd., Chennai, India Printed in Great Britain on acid-free paper by Biddles Ltd, www.biddles.co.uk ISBN 978–0–19–852543–1 (Hbk.) ISBN 978–0–19–921918–6 (Pbk.) 1 3 5 7 9 10 8 6 4 2

Preface

Almost every month some book or television programme describes exciting developments in cosmology or fundamental physics. Many tell us that we are on the verge of finding the explanation for the Big Bang or the ultimate Theory of Everything. These will explain all physics in one fundamental set of mathematical equations. It is easy to be swept along by the obvious enthusiasm of the participants, particularly when they are making real progress in pushing back the boundaries of knowledge. Unfortunately, most of their brilliant new ideas are doomed to be forgotten, if only because they cannot all be right. Consider the currently fashionable idea that our universe is just one of many unobservable, parallel universes, all equally real. How can one hope to describe the inner structures of such universes, each with its own values of the ‘fundamental’ constants? Many may be dull and featureless, but others are presumably as fascinating and complex as our own. However much some physicists declare the reality of these other universes, in practice their main function is to support the mathematical models of the day, or to ‘explain’ certain properties of our own universe. My goal in this book is not to adjudicate on the correctness of such new and speculative theories. We will instead consider the development of science in a historical context, in order to find out how such questions have been resolved in the past, and to explain why many long established ‘facts’ have turned out not to be so certain. My conclusion is surprising, particularly coming from a mathematician. In spite of the fact that highly mathematical theories often provide very accurate predictions, we should not, on that account, think that such theories are true or that Nature is governed by mathematics. In fact the scientific theories most likely to be around in a thousand years’ time are those which are the least mathematical—for example evolution, plate tectonics, and the existence of atoms. The entire book is effectively an extended defence of the above statements. In the course of the discussion I risk the displeasure of many of my colleagues by explaining the feebleness of mathematical Platonism as a philosophy. I also provide psychological and historical support for the claim that mathematics is a human creation. Its success in explaining nature is a result of the fact that we developed much of it for precisely that purpose. Even the numbers which we use in counting become no more than formal symbols, invented by us, as soon

vi Preface

as they are as big as 101000 (1 followed by a thousand zeros). Pretending that we can count from 1 up to such a number ‘in principle’ is a fantasy, and will always remain so. Moreover, it is not necessary to believe this in order to be interested in pure mathematics. Whatever some over-enthusiastic physicists might claim, there is much which is beyond our grasp, and which will probably remain so. Subjective (first person) consciousness is one such issue. Understanding the true nature of quantum particles is another, in spite of the proven success of the mathematical aspects of quantum theory. Contingency, or historical accident, has obviously had a major influence on geology and biology, but some physicists think that it is even involved in the form of the laws of physics. Whether or not this is true, scientists are right to believe that, with enough effort, they can push the boundaries of their subjects far beyond their present limits. An unusual feature of the book is that I try to explain why philosophical issues are important in science by means of simple examples. This is not the style followed by academic philosophers, but it makes the issues easier to understand, particularly in a popular context. In addition, discussions about the status of money, zombies, or rainbows are more fun than dry logical arguments about ontology. I am painfully aware that the scope of the book is far wider than anybody’s expertise could span in this age of specialists. The attempt is worth making, because arguments informed by only one branch of science are inevitably distorted by that fact. I do not claim to have found the final answer to all of the deep questions in the philosophy of science, but hope that readers who have not previously thought much about these will see why they are important. People vary enormously in their liking of mathematics. Many switch off as soon as they see it, and editors of popular books advise their authors to reduce it to the absolute minimum. I have gone as far as I can in this direction, and reassure the allergic reader that any difficult passages can be skimmed over. They are present to ensure that interested readers do not feel cheated by being told conclusions without any evidence in their support. I wish to acknowledge invaluable advice, or sometimes just stimulation, which I have received from many friends and colleagues, in particular Martin Berry, Alan Cook, Richard Davies, Donald Gillies, Nicholas Green, Andreas Hinz, Hubert Kalf, Mike Lambrou, Peter Palmer, Roger Penrose, David Robinson, Peter Saunders, Ray Streater, John Taylor and Phil Whitfield. I do not, however, burden them with the responsibility of agreeing with anything I say here. I also thank my family for providing an atmosphere in which a task such as this could be contemplated; I know that the time which I have devoted to it has put me in great debt to them.

Contents

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Perception and Language 1.1 Preamble 1.2 Light and Vision Introduction The Perception of Colour Interpretation and Illusion Disorders of the Brain The World of a Bat What Do We See? 1.3 Language Physiological Aspects of Language Social Aspects of Language Objects, Concepts, and Existence Numbers as Social Constructs Notes and References

1 1 3 3 4 6 13 15 16 18 18 22 24 27 31

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Theories of the Mind 2.1 Preamble 2.2 Mind-Body Dualism Plato Mathematical Platonism The Rotation of Triangles Descartes and Dualism Dualism in Society 2.3 Varieties of Consciousness Can Computers Be Conscious? Gödel and Penrose Discussion Notes and References

33 33 34 34 37 41 43 46 49 50 52 54 59

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Arithmetic Introduction Whole Numbers

61 61 62

viii Contents

Small Numbers Medium Numbers Large Numbers What Do Large Numbers Represent? Addition Multiplication Inaccessible and Huge Numbers Peano’s Postulates Infinity Discussion Notes and References

62 64 65 66 67 68 71 75 78 80 83

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How Hard can Problems Get? Introduction The Four Colour Problem Goldbach’s Conjecture Fermat’s Last Theorem Finite Simple Groups A Practically Insoluble Problem Algorithms How to Handle Hard Problems Notes and References

85 85 87 88 89 90 91 93 96 97

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Pure Mathematics 5.1 Introduction 5.2 Origins Greek Mathematics The Invention of Algebra The Axiomatic Revolution Projective Geometry 5.3 The Search for Foundations 5.4 Against Foundations Empiricism in Mathematics From Babbage to Turing Finite Computing Machines Passage to the Infinite Are Humans Logical? 5.5 The Real Number System A Brief History What is Equality? Constructive Analysis Non-standard Analysis

99 99 100 100 103 103 107 109 113 116 117 123 125 127 130 131 134 135 137

Contents ix

5.6 The Computer Revolution Discussion Notes and References

138 139 140

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Mechanics and Astronomy 6.1 Seventeenth Century Astronomy Galileo Kepler Newton The Law of Universal Gravitation 6.2 Laplace and Determinism Chaos in the Solar System Hyperion Molecular Chaos A Trip to Infinity The Theory of Relativity 6.3 Discussion Notes and References

143 143 146 151 153 154 157 158 160 161 163 164 166 170

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Probability and Quantum Theory 7.1 The Theory of Probability Kolmogorov’s Axioms Disaster Planning The Paradox of the Children The Letter Paradox The Three Door Paradox The National Lottery Probabilistic Proofs What is a Random Number? Bubbles and Foams Kolmogorov Complexity 7.2 Quantum Theory History of Atomic Theory The Key Enigma Quantum Probability Quantum Particles The Three Aspects of Quantum Theory Quantum Modelling Measuring Atomic Energy Levels The EPR Paradox Reflections Schrödinger’s Cat Notes and References

171 171 172 174 175 175 176 177 178 179 181 182 183 184 186 188 190 192 193 195 196 198 199 202

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8 Is Evolution a Theory? Introduction The Public Perception The Geological Record Dating Techniques The Mechanisms of Inheritance Theories of Evolution Some Common Objections Discussion Notes and References

203 203 204 205 209 213 217 225 230 232

9 Against Reductionism Introduction Biochemistry and Cell Physiology Prediction or Explanation Money Information and Complexity Subjective Consciousness The Chinese Room Zombies and Related Issues A Physicalist View Notes and References

235 235 238 240 242 243 245 246 248 250 251

10 Some Final Thoughts Order and Chaos Anthropic Principles From Hume to Popper Empiricism versus Realism The Sociology of Science Science and Technology Conclusions Notes and References

253 253 256 259 266 270 274 276 279

Bibliography

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Index

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1 Perception and Language

1.1

Preamble

Most of the time most people relate to the world in a pretty straightforward way. We assume that entities which appear to exist actually do so, and expect scientists to provide us with steadily more detailed descriptions of their underlying structures. We try not to worry about the fact that fundamental theories are highly mathematical, and hence incomprehensible to almost everyone. Some, such as the Oxford chemist Peter Atkins, find the prospect of ultimately explaining the whole of reality in mathematical terms exhilarating, while others fear or reject it because of its impersonal character. There are a few puzzles associated with this scientific picture of reality. One is the nature of subjective consciousness, which used to be called the human soul, and which some philosophers now regard as an illusion. Another is the status of mathematics: why should the ultimate explanation of reality be in terms of equations? Roger Penrose has addressed these fundamental questions in his books The Emperor’s New Mind and Shadows of the Mind, published in 1989 and 1994 respectively. Roger is an outstanding mathematical physicist, but I think that his approach to these issues is quite wrong, and in this book I propose an entirely different way of looking at them. Readers will probably be relieved to hear that they are not going to be asked to wade through page after page of detailed mathematics or logic. Although it contains some mathematical results as illustrations, this book does not involve any deep technical arguments. One of Penrose’s principal ideas is that Gödel’s theorems, discussed on page 111, prove that human beings can understand results which are beyond the capacity of any computer. He believes that they also provide a route by means of which one can understand the mathematical mind, and by extension the nature of consciousness. This is pretty optimistic, to say the least. Penrose makes strong statements about the limitations of computers, but ignores the obvious fact that the human mind also has limits. Mathematics provides one of the last refuges of Platonism, discussed in some detail on pages 27 and 37. I will argue that this philosophy is entirely unhelpful in understanding either mathematics or its relationship with the

2 Preamble

outside world. The high degree of abstractness of the subject is shared by chess, philosophy, and music, and does not require any special explanation. I thus reject the Platonistic position of a sizeable fraction of my colleagues, including some of the most eminent. On the other hand, the ideas presented here are entirely in line with modern experimental psychology and the history of mathematics itself. Re-establishing the links between mathematics, science, and other human concerns involves a rejection of the ‘easy’ reductionist option, which leaves subjective consciousness out in the cold. This book does not provide the solution to every problem about the nature of reality, but presents a series of arguments suggesting that we must stop looking in directions which leave us out of the picture. Platonism, in which mathematics exists in some ideal world unrelated to human society, is a typical example of this. Since the time of Descartes, Western science has developed along a route which has been immensely successful for those aspects of reality in which human issues are of little relevance. Its very success has encouraged scientists to avert their gaze from those aspects of reality which their methods say little about. Some have even convinced themselves that there are no such aspects. In this chapter we consider the evidence that almost everything relating to human knowledge is more problematical that we normally admit. We start with a review of recent work in experimental psychology, because it is surely necessary to understand our physical nature if we are to understand the nature of our thoughts. This chapter is absolutely mainstream psychology. I cannot make quite the same claim about Chapter 2, because most deep questions in philosophy remain controversial. From Chapter 3 onwards we will cover a wide range of sciences, indeed any area in which there is controversy about the bases for claims of objective knowledge. The first half of this chapter describes the wide variety of methods which have been used to investigate the differences between what we think we see and reality itself. Not only have these investigations provided a consistent description of the world, but they even explain why our unaided senses paint a distorted, indeed different, picture. Particularly important in this respect has been the development of brain scanning machines, which are beginning to give detailed information about what is happening as our brains struggle to interpret sensory data. This is one of the most exciting current fields of scientific research. As a society we are progressively re-adjusting our world-view in the direction indicated by our instruments and intellects. To give just one example: we commonly talk about a ‘fluid’ called electricity which can flow through solid copper wires but not through the open air; this fluid can be stored in batteries, even though a full battery looks the same and is no heavier than an empty one.1 We accept such bizarre propositions in spite of a complete lack of direct sensory evidence because they provide consistent explanations of observed phenomena, such as the fact that a light bulb becomes bright when we turn a switch. For the first time in history large parts of our lives depend upon

Perception and Language 3

machines and ideas which would appear magical or incomprehensible to our ancestors. In the second half of the chapter we discuss the relationship between language and reality, which turns out to be a much harder task.

1.2

Light and Vision

Introduction The view that our senses provide us with direct and straightforward information about the outside world was promulgated by Aristotle, St. Thomas Aquinas, and then by the sixteenth century scholastic philosophers. The first person to criticize it systematically was Descartes, whose philosophical and scientific ideas will be discussed in more detail in Chapter 2. In Le Monde, 1632 he wrote: In proposing to treat here of light, the first thing I want to make clear to you is that there can be a difference between our sensation of light . . . and what is in the objects that produces that sensation in us . . . For, even though everyone is commonly persuaded that the ideas that are the objects of our thought are wholly like the objects from which they proceed, I see no reasoning that assures us that this is the case.

Newton later provided positive reasons, described below, for distinguishing between colours and our sensations of them, and these have been reinforced by all recent psychological research. Our present understanding of brain function has involved many different lines of investigation. One is the study of optical illusions, which provide hints about the brain mechanisms involved in ‘normal’ vision. Secondly, psychologists study the abnormal thought processes of people who have suffered specific brain damage; this helps them to discover which regions of the brain are involved in different types of processing. There has been extensive analysis of the biochemistry and structure of individual nerve cells, and of the anatomy of the retina and the rest of the brain. Another rapidly developing field of psychological research depends upon the use of brain scanning machines: these can identify which parts of the brain are most active when people are asked to carry out various mental tasks. Research in each of these fields forces us to the conclusion that the unconscious part of our brain constructs the reality in which we live; evolution has seen to it that these mental constructions lead to appropriate behaviour in most normal circumstances. Donald Hoffman gives a clear statement of this conclusion from the point of view of an experimental psychologist in Visual Intelligence: How We Create What We See. He explains why it is possible for us all to agree about the nature of the world and nevertheless for us to be fundamentally wrong in the way we see it. Subjective pictures are not just part of picture perception. They are part of ordinary everyday seeing. And that should come as no surprise. You construct

4 Light and Vision every figure you see. So, in this sense, every figure you see is subjective. . . . But then why do we all see the same thing? Is the consensus magic? No. We have consensus because we all have the same rules of construction.

According to Hoffman the rules of construction are built into the anatomy of our brains, and cannot be modified by the exercise of rational thought. Lest you think that this is just Hoffman’s personal view, let me quote a corresponding passage from Francis Crick’s The Astonishing Hypothesis. What you see is not what is really there; it is what your brain believes is there. In many cases this will indeed correspond well with characteristics of the visual world before you, but in some cases your ‘beliefs’ may be wrong. Seeing is an active constructive process. You brain makes the best interpretation it can according to its previous experience and the limited and ambiguous information provided by your eyes.

These ideas seem rather disturbing, but would have been regarded as absolutely orthodox Taoist philosophy in tenth century China. The book Hua Shu of this period describes a kind of subjective realism, in which the external world is real, but our knowledge of it is deeply affected by the way in which it is perceived, so that we cannot seize its full reality. Like Hoffman and Crick, the (supposed) author T’an Ch’iao even refers to optical illusions and human inattention to press the view that we pick out certain elements of reality to form our world-picture.2 The ideas above provide strong warnings against believing that something is true simply because it matches our intuition well. We can gain objective knowledge about the underlying reality, but this depends upon learning to accept the verdict of our instruments rather than of our unaided senses. We have chosen this path because such a wide variety of different methods of scientific investigation have led to a consistent picture. Indeed they even explain why the evidence of our own senses is not a reliable guide to the nature of reality.

The Perception of Colour The study of optical phenomena was slow to develop historically because of the great difficulty of disentangling the physical, physiological, and psychological aspects of the subject. It provides a very clear example of the immense gap between our perceptions and the physical reality which lies behind them. Although the Pythagoreans maintained that light travelled from the eye to the object, Lucretius got much closer to the truth in The Nature of the Universe: No matter how suddenly or at what time you set any object in front of a mirror, an image appears. From this you may infer that the surfaces of objects emit a ceaseless stream of flimsy tissues and filmy shapes. Therefore a great many films are generated in a brief space of time, so that their origins may rightly be described as instantaneous. Just as a great many particles of light must be emitted in a brief space of time by the sun to keep the world continually filled with it, so objects in general must correspondingly send off a great many images in a great many ways from every surface and in all directions simultaneously.

Perception and Language 5

Lucretius also argued that the colours of objects were not intrinsic to them, since the sea could have a variety of appearances according to the way that its component atoms were churning around inside it. In spite of the startling accuracy of these ideas for the time, they cannot be classified as science: Lucretius could propose no quantitative tests of his ideas, which were, eventually, just speculation. The scientific investigation of light and colour started in the seventeenth century, with major investigations by Robert Hooke, Christiaan Huygens, and Isaac Newton. Newton used prisms to split white light into its component colours, and then recombined the components back into white light; this led him to understand that white light was not pure, as seems naively obvious, but a mixture. He understood the cause of chromatic aberration in the lenses of refracting telescopes, and designed and built the first reflecting telescope in 1669. His paper Theory of Light and Colours, published in 1672, attracted great attention and also started a feud between him and Hooke. They differed sharply about whether light should be regarded as corpuscular or wave-like, a debate which was to continue until the twentieth century, when quantum mechanics allowed it to be both. We now know that light comes in a continuous range of wavelengths, and that our eyes are only sensitive to a very narrow band of these. Our colour discrimination depends upon our having three kinds of receptor, called R, G, and B cones, in our retinas, each of which is most sensitive to a particular range of wavelengths. These receptors cannot possibly distinguish between all the wavelengths in visible light, so what we see is a great simplification of what is in the light itself. Objects are not red, green, or blue in themselves: our impressions are created by neural processing of the very limited information provided by our retinas. People actually have one of two types of R cones, which are genetically inherited. These produce slight differences in perception between individuals, which may be important when matching colours. The variation is caused by a single amino acid change in the relevant protein, and provides a rare instance in which we know the precise causal chain from a change at the molecular level to a difference between the subjective worlds two people may inhabit.3 This provides a partial answer to the philosophical question of how we can know that two normal people have the same subjective colour experiences: they need not. This is not merely an abstract problem. I myself have had regular disagreements over many years with my wife about the nature of colours on the borderline between green and blue. We cannot even agree whether this is a difference of naming or of perception. Maybe our colour receptors are indeed slightly different, and we have been taught to name colours by parents who had the same types of receptor as ourselves. These small differences pale into insignificance once one compares our visual experiences with those of other species. It is known that many birds and insects are sensitive to ultraviolet light. Ultraviolet photographs of some flowers reveal patterns, invisible to us, which are important to insects seeking nectar.

6 Light and Vision

On the other hand most mammals have only two kinds of colour receptor. Our very similar R and G cones appear to have evolved from an earlier single type recently, possibly to improve our ability to discriminate between fruits.4 In the most common form of colour blindness, affecting very roughly 5% of males, either the R or the G cones are missing, so the person cannot distinguish red from green. We regard such people as having a disability, but by the standards of most mammals they are normal. On the other hand pigeons have six or more different types of colour receptor, and might regard all humans as having only partial colour vision! We conclude that the same light falling on the eyes of different species must produce very different subjective colour impressions in their brains. Returning to human beings, it is known that quite different combinations of wavelengths may produce the same subjective impression. Whether the names of colours are simply cultural constructs or have a physiological basis is again a matter of active debate. The comparative study of a large number of languages shows that although they may have different numbers of named colours these are classified into a coherent hierarchy. Namely if any colour in the box below appears in a language then all of the colours on previous lines also appear. This suggests that there is a physiological basis for the existence of colour names, even if there is no external physical basis. Unfortunately, even this conclusion has recently been thrown into doubt by a study of the Burinmo tribe in Papua New Guinea. The colour names of this tribe are radically different from the below list and their ability to distinguish colours is positively correlated to their colour language. These observations do not support the idea that colour categories could be universal.5 In biology almost everything is more complicated than initial analyses suggest!

white, black red green, yellow blue brown purple, pink, etc.

Interpretation and Illusion There are many other differences between our perceptions and the reality behind them. When light falls on a retinal receptor, it emits pulses which are then processed in stages, first in the retina and then in the brain. Each level of processing involves further interpretation and selection, all of which happens before we become conscious of the scene before us. In most cases we are

Perception and Language 7

unaware that these processes are going on, but it is possible to set up situations in which we can see that our lower level interpretations are quite incorrect. Understanding the way in which images are processed is a research field of great complexity, and my goal here is simply to draw attention to the variety of mechanisms involved, and a few of the ways in which they can fail. When this happens we experience an optical illusion. A simple example, much exploited by Bridget Riley and other artists in the Op Art movement of the 1960s, involves a property of our peripheral vision. The phenomenon can be seen by moving your head towards and away from figure 1.1, while concentrating on the spot in the centre. The strong sense that the rings are rotating, in opposite directions, depends upon the fact that the peripheral part of our retina is primarily concerned with the detection of movement. The neural circuits involved are designed, for obvious evolutionary reasons, to ‘fail-safe’: it does not matter too much if a non-existent movement is reported, but might be fatal if an actual movement is missed, even once. Even when we recognize that the effect is illusory, we cannot prevent it happening, because the neural processing happens below the level at which we have conscious control. Judging the brightness of a part of a picture is not a simple matter. In figure 1.2, drawn by Ted Adelson, the two squares labelled A and B are exactly the same shade of grey. This may be checked by covering up everything in the picture except these two squares. The reason for the illusion is that your visual system is not interested in the true luminosity of the squares. One part interprets the picture as being of a three-dimensional object, and passes this conclusion to

Fig. 1.1 Rotating Rings

8 Light and Vision

Fig. 1.2 Checker-shadow Illusion Reproduced by permission of Edward H. Adelson, Department of Brain and Cognitive Science, MIT

another part, which compensates for what it considers to be the likely variations of lighting. By the time you become conscious of the picture, these adjustments are simply a part of what you see. The extent to which ‘seeing’ depends upon active brain processes became very clear to me on a recent holiday in Madeira. Standing on the edge of a shallow pool one sunny day, a companion remarked on the number of small fish in it. Although I looked hard through the constantly varying pattern of surface ripples I could not see any. My companion explained carefully what I should look for and within a minute or so my brain reprogrammed itself, and hundreds of the fish became clearly visible. Indeed I could hardly understand how I had not been able to see them before. This is not an isolated example. So-called ‘primitive’ peoples learn to recognize myriads of subtle features of their environments which urban travellers are completely unaware of. These may be vital for avoiding dangers as well as for finding sources of food. Figure 1.3 ‘Random Points’ shows how powerful the mechanisms involved are. As soon as you are told that there is one ‘extra’ point in the figure, you can identify it as one out of two possibilities without consciously looking at most of them. This feat can only be achieved so quickly because your visual system processes the whole picture simultaneously. In computer terms it is a massively parallel system. Fortunately such tasks do not

Perception and Language 9

Fig. 1.3 Random Points

need to be carried out using our rational faculties, which would be much less competent at such tasks! Let us turn to the way in which we construct three-dimensionality from what we see. Following the re-discovery and elucidation of the laws of perspective by Brunelleschi and Alberti in the first half of the fifteenth century, Hogarth was one of the first painters to produce pictures with deliberately impossible perspectives. In fact it is embarrassingly easy to follow the laws of perspective rigorously, while producing impossible objects, as figure 1.4 shows.

Fig. 1.4 Part of a Fence

10 Light and Vision

Similar ideas underlie several of the paintings of M. C. Escher, such as Ascending and Descending, 1960, in which a chain of monks climb a staircase which apparently returns to its starting point, even though every step is upwards. Escher cleverly incorporated enough distractions into the picture that it does not appear particularly strange to the eye. These illusions are possible because our visual system has to make guesses based on incomplete information. It is a fact that if an object exists then a drawing of it will follow the laws of perspective. However, our visual system follows an incorrect rule: that if a drawing follows the laws of perspective then a corresponding object exists, or could exist. It is worth mentioning that the issue of interpretation is one of the barriers to developing the ability to draw faces: untrained people draw their interpretation and not what they see, with the result that they can draw a face more accurately if it is presented upside down. Vivid evidence of the brain’s construction of images is provided by autostereograms, one of which is shown in figure 1.5. At first sight a random collection of dots, if you focus on a point behind the image, after a period of up to a minute a three-dimensional picture of an oval with a square hole should emerge.6 The effect depends upon the fact that we have two eyes, which can be persuaded to look at different parts of the autostereogram. The following experiment is well worth trying. Get a small piece of card and hold it close to your face and slightly to the side of one eye while you look at the autostereogram. Now move the card slowly until it partly covers one pupil. The result is that the part of the picture which is only seen by one eye returns to its random appearance while the part still visible to both eyes retains the three-dimensional image. Nevertheless both parts are equally clear. This is a particularly effective way of isolating the part of the visual system which constructs three-dimensional effects. The information needed to construct the three-dimensional picture is of course in the autostereogram, but the picture itself is not. When we look at the world our brains decide that some objects are stationary in spite of the fact that as we look at different parts of them, the image on our retina is continually changing. Unless we are almost asleep, our brain factors such changes out before the mental image reaches our consciousness, informing us only of its current conclusions about which of the objects seen are stationary and which moving. Our ‘mental world’ is quite distinct from the constantly moving image on our retinas. The compensation mechanism is very specific and fails if one closes or covers one eye and presses the other eyelid gently from the side. Presumably the reason for this is that there has been no need for our brains to take into account the possibility of visual changes caused by pressing eyelids! The above are a tiny fraction of the interesting ideas in this rapidly developing field. We have not listed the thirty-five specific rules of visual interpretation which Hoffman describes. These control what we think we see, which may or may not be correct in particular circumstances. We should not be surprised about this: natural selection worked to ensure that in the kind of circumstances

Fig. 1.5 Oval with Square Hole Drawn using Randot vl.1 software written by Geoffrey Hausheer

12 Light and Vision

we evolved in, our responses to visual stimuli normally promote our survival. It did not work to ensure that in the very specific situations dreamed up by psychologists the interpretations should bear any relationship with the truth. The fact that we recognize something as an illusion created by fallible brain machinery does not enable us to banish the mistaken impression. Of course, given our intellect, we can often compensate for the mistake in a way which other animals almost surely cannot. But Hoffman points out that the situation is worse than this. Certain aspects of our visual interpretation are so deep-seated that we can hardly conceive that our mental constructions of the objects are quite distinct from the objects as they really are. It is necessarily difficult to expand on this idea, but he describes the analogy of computer games with multiple human players. The people involved have the feeling that they are carrying out actions in a virtual landscape, and it is clear that the interactions between the players have an objective aspect: different players agree about the progress and outcome of the game. On the other hand, what is actually happening can only be explained in terms of a collection of electrical currents flowing through circuits inside several computers. So the mental experience is caused by an artificial system whose nature is entirely unperceived by the participant. It could be argued that the fact that we have rules of interpretation and that we may be led into error in some contrived situations has no philosophical importance: in all normal situations, if we have a subjective impression of a table in front of us, that is because there is a table in front of us, and this is what constitutes seeing the table. On the other hand, one does not need to be a philosopher to appreciate that we are only aware of the surface appearance of the table, and occasionally of its weight. The manufacturers of rosewood tables exploit this by restricting the rosewood to a thin surface veneer. If our sense organs enabled us to ‘see’ the interior of tables, this cost-saving device would fail utterly. There are quite ordinary situations in which what we see has an obviously uneasy relationship with what is there, the most obvious being when we look at a mirror. The image we see seems to be behind the glass, but we interpret it as being a reflection. Our ability to make this interpretation is shared by very few other mammals, even though their eyes have similar structures to our own. Even we occasionally find it hard to relate to this image: when I was younger I made frequent efforts to cut my hair in a mirror, but never really mastered the skill. A person who could not recognize himself in a mirror would be abnormal by our standards, but would be no worse off than most animals. We, in turn, would be regarded as grossly mentally deficient by an alien which could cut its hair in a mirror without effort, which recognized faces upside down as easily as if they were the right way up, or which could ‘see’ the route through a complicated maze drawn on paper without conscious effort. What seems straightforward and obvious is, in fact, highly species-dependent. It depends entirely upon what unconscious processes your brain is capable of carrying out.

Perception and Language 13

It might nevertheless be said that in the above case one sees an image ‘of oneself’ in the mirror, and it is related in an objective fashion to how one actually is, so one does indeed see oneself. Now imagine a future world in which all public advertisements make use of holograms, and in which television newscasters are computer simulations of people who are long dead or never existed. Using technology which almost exists today we may be surrounded by images which are not based upon any real object. We would be seeing something, but it would not be what it seems to be, nor would anyone think that there should exist any objects relating to what they see in everyday life. Some of the above examples might seem to be frivolous, but when we get to the discussion of quantum theory we will confront the possibility that our brains may not be capable of constructing any comprehensive visual model of what is going on. The quantum world is really and objectively there, but it is so remote from the world in which we have evolved that we may never be able to construct an intuitive model of it. Almost every physicist agrees that the real nature of quantum particles remains beyond our imagination, and most probably agree that the only comprehensive model we will ever have of quantum theory will be a purely mathematical one.

Disorders of the Brain The last section concentrated on the normal properties of the visual system, but there are many perceptual abnormalities (agnosias) which result from damage to particular parts of the brain. These further demonstrate the extent to which our view of the world depends upon interpretation within the brain. One of these, called cinematic vision, occurs when a person with perfectly clear eyesight is unable to recognize motion. The person afflicted sees a series of still views of objects, so that a car approaching is seen first as a small vehicle in the distance and then suddenly as a much larger one close up. Similarly a sufferer trying to pour a cup of tea may first see a static tube joining the teapot to the cup, and then suddenly a large pool of tea covering the table. In blindsight a person is not consciously aware of objects in a certain part of the field of vision, even though their eyes are perfectly normal. When asked to guess what is present, and where it is, they are frequently correct, to their own surprise. Very recently brain scanners have provided evidence that images on the ‘blind’ side of the field of vision are processed differently from those on the normal side; the method of processing presumably bypasses whatever brings the perceptions to the consciousness of the person. These fascinating discoveries have the potential of providing deep new insights into the nature of the ‘consciousness mechanism’ in the brain, and are the subject of active research. The term recognition agnosia refers to the inability to recognize an object by sight even when it can be recognized easily by touch, or the inability to recognize the faces of close friends and family even though their voices evoke

14 Light and Vision

normal responses, or the inability/refusal to recognize that one side of the body actually belongs to the sufferer. In 1985 Oliver Sacks described a patient who was a talented musician and able to engage normally in conversations in spite of the fact that he was unable to recognize most common objects visually. Particularly strange was that he seemed to accept his failure to recognize, say, a rose or a glove visually as entirely unremarkable, when he could give accurate descriptions of their parts and colours. This kind of mental loss is much more disruptive of normal life than would be the loss of vision, since it involves the partial disintegration of the personality. Hirstein and Ramachandran have recently made an in depth study of a man who developed Capgras syndrome following a head injury.7 Tests showed that he had no obvious deficits in higher functions and no evidence of dementia, in spite of the fact that he believed that close family members were impostors who looked exactly like the genuine people. Indeed he suffered the same problem with respect to himself. He recognized mirror images as being of himself, but would refer to photographs as being of another person who looked exactly similar; he sometimes even referred to himself as not being the genuine person. The best explanation of this syndrome at present is that there are two separate circuits involved in relating to close relatives, one dealing with recognition and the other creating an appropriate emotional response. If the circuit producing the emotional response does not function then it may be impossible for the unfortunate person to believe that the relative is who they seem to be. The fact that this may even apply to the person’s response to himself raises a deep philosophical question about the nature of our self-consciousness. It appears that even this is not a unitary entity, but involves the correct interaction of a variety of independent modules. A provocative way of putting it is that our sense of self is created by the modules in our brains in order to help it to function. Turning to mathematics, it has become clear that the ability to distinguish between very small numbers, those below about 4 or 5, does not involve counting but depends upon a specific module, probably in the left inferior parietal lobe. In The Mathematical Brain Brian Butterworth emphasizes that reasoning about even very small numbers involves a specific mechanism. People whose number module is damaged, either from birth or because of a stroke, may have perfectly normal intelligence apart from the fact that they have serious deficiencies in any situations which involve even very small numbers. Some cannot see without counting that a group of three objects is bigger than a group of two similar objects. By timing how long they take to do simple comparison tasks, it has been discovered that they may find it as hard to distinguish between the pair 9, 2 as between the pair 9, 8. Such people either cannot cope with numbers bigger than 5 by formal counting, because they do not understand what counting means in its application to the real world, or they count very slowly and painfully and only up to rather small numbers. This problem is now a recognized disability called dyscalculia, and is sometimes associated with dyslexia. We like to believe that many of the skills mentioned in the paragraphs above are matters of general intelligence, but they are not, and this must undermine

Perception and Language 15

the classical view of consciousness and rationality as unified entities. If we act rationally in some situation this is because each of the modules in our brain behaves appropriately in that situation. This being so, one can imagine our distant descendants possessing extra modules in their brains whose function we are incapable of comprehending, and which enable them to understand matters entirely beyond our mental grasp. Our invention of language and subsequently of science have enabled us to progress far beyond what our unaided minds can grasp, and have led us into territories which we could never have entered without their support, but there are still limits to our mental capacities. Some indications of the extent of these will be presented in later chapters.

The World of a Bat It is well known that the vision of frogs is dramatically different from ours. They do not see static objects, and can only react to motion. As a result, if surrounded by recently killed insects they will starve, but as soon as one flies across their field of vision they can react appropriately. In this section, however, we will discuss bats, because their quite different type of perception cannot be so easily dismissed as just an inferior version of our own. When we consider the perception of bats below, we will be referring to their echo location system, and not their vision. Because bats emit high pitched series of clicks and are aware of the time delay and pitch of the echoes, they have precise information about the distance and rate of movement of obstacles or prey. This has some quite important implications for their perception of the world. The first is that distant objects must appear much darker, or dimmer, to them than closer objects, because the intensity of the echo from an object decreases very rapidly with its distance. For humans the apparent brightness of an object stays the same as it moves away, and only its size decreases. More importantly it is likely that a (hypothetical intelligent) bat would not consider that a picture of an object has any similarity to the object itself. Since its radar builds in distance information, a picture must appear to a bat to be a flat pigmented rectangle, quite unlike the three-dimensional object which it seems to resemble in our minds. We appreciate flat pictures because our vision is essentially two-dimensional, but the bat would be correct in maintaining that there is no physical similarity. We cannot really know what subjective impressions bats experience, but the following thought experiment may help. Let us try to imagine what vision would be like in a world in which green light moved through the air much more slowly than red light. When viewing a static object the time delay for the arrival of green light compared with red would make no difference to our perceptions. Now suppose the object starts to move to the right. The red image emanating from the object at any moment reaches our eyes slightly earlier than the green image produced at the same moment—correspondingly, at any moment we see a red image which was produced at a slightly later time than that at which

16 Light and Vision

Fig. 1.6 Colour Fringes

the green image was produced. The result is that the object acquires a red fringe on its right side and a green fringe on its left side, as in figure 1.6—in which the colour fringes are replaced by hatching. Since we could already see how fast the object was moving, we could use this effect to draw conclusions about its distance: the further away it was the thicker the fringes would be. Let us now imagine that a module in our brains could interpret the colour fringes before they reached the conscious mind. Then we might have an enhanced threedimensional perception of objects, but only if they were moving across the field of view. Finally suppose that the object is moving straight towards us. Then its boundaries are expanding on all sides, so it will be completely surrounded by a red fringe, and once again we might be able to perceive its rate of approach to us particularly clearly while it remains moving. These extra senses would be extremely valuable in a society which is so heavily dependent on cars. Suppose instead that the S (blue) colour receptors in our retina responded not to the colour of light but to the distance of the object being viewed, while everything else about our colour vision was unchanged. Then we would look around and see objects with various shades and combinations of colours as at present. However, we would know that the more blue an object was the closer it was. This would provide a much enhanced sense of depth. It might be possible to implement this idea using modern computer processing and virtual reality displays, and it might even be useful to people such as pilots of aircraft. Perhaps this idea has already been patented!

What Do We See? In the early days of research on vision, it was believed that the image falling on the retina was mapped with some modifications onto a part of the brain, where our mind became conscious of it. This led to the joke about a homunculus inside the brain ‘looking’ at the image laid out somewhere there. As a result of years of experimentation we now have a very different picture. The image falling on the retina is torn into fragments, so that edges,

Perception and Language 17

colours, motion, and particular shapes such as mouths and eyes are all analysed separately. There are even specialized neural circuits which detect only edges with particular orientations. At the end of this process a new and quite different ‘image’ is constructed, which we commonly suppose to be a ‘true’ representation of the original three-dimensional object. If any of the separate modules which process different aspects of the original image is damaged by a stroke, or functions incorrectly because the image is highly unnatural, then we get at best an optical illusion and at worst a completely incomprehensible result. According to experimental psychologists, our subjective impression is never of the object as it is. It is a construction which enables us to behave appropriately in almost all ordinary circumstances. Evolution has ensured that our constructions give us a useful picture of reality, one which generally helps us to survive. These experimental findings should encourage us to re-examine the way in which we relate to our everyday surroundings. People rarely think about the extent to which we are obsessed by the surfaces of objects. Objects are threedimensional and most of their material is inside them, not on their surface. How many of us ever think in a tactile as opposed to an intellectual manner about the thousands of kilometres of ground underneath us? The existence of these things is known rationally, but our senses do not inform us about them, so we ignore them. Presumably cows have no concept that there might be anything underneath the earth and grass they stand on, even though their vision is quite similar to our own. On the other hand those of us who live in the countryside often contemplate the stars in the night sky, which are far more remote, simply because our senses do inform us about their existence. To the extent that we have a correct or true view of reality it is a result of the use of our intellects rather than simply because of the evidence of our senses. Over many centuries we have learned that the Sun is stationary although it seems to move, and the Earth rotates although it appears to be stationary. We have learned that a table is almost entirely composed of discrete atomic nuclei and electrons separated by empty space, although it appears to be solid and continuous. We have come to accept that TV programmes can travel through empty space to our sets even though our senses provide no direct evidence of this. We devote enormous technological resources to the avoidance of infections by invisible particles called bacteria and viruses. These facts, and many others, show how heavily our interpretation of reality depends upon the technical knowledge accumulated by the society we are born into. The idea that our instruments provide a truer picture of reality than our senses arose in the seventeenth century. It was a key ingredient in the scientific revolution, to be discussed in Chapter 6. Robert Hooke expressed it as follows in Micrographia, published in 1665: The next care to be taken, in respect of the Senses, is a supplying of their infirmities with Instruments, and, as it were, the adding of artificial Organs to the natural; this in one of them has been of late years accomplisht with prodigious benefit to all sorts of useful knowledge, by the invention of Optical

18 Language Glasses . . . It seems not improbable, but that by these helps the subtilty of the composition of Bodies, the structure of their parts, the various texture of their matter, the instruments and manner of their inward motions, and all the other possible appearances of things, may come to be more fully discovered; all which the ancient Peripateticks were content to comprehend in two general and (unless further explain’d) useless words of ‘Matter’ and ‘Form’.

The main change since Hooke wrote those words is that he thought primarily in terms of augmentations of existing senses, whereas modern instruments provide us with ‘senses’ quite unlike any which we naturally possess. Our new reliance upon instruments is not as straightforward as it appears. They do not tell us anything about reality until we interpret the readings we obtain from them in the light of some theory of how they work. We are convinced that this is not a circular process by the huge variety of independent sources of confirmation of the picture which we have built up over centuries of scientific investigation. I shall have more to say about this in Chapter 10.

1.3

Language

Physiological Aspects of Language The visual system of humans is amazingly sophisticated, but it is not radically different from that of other mammals. Many experts consider that the best bet is that our specifically human intelligence is related to our use of language. Although language is clearly very important, it is easy to be carried away by this line of argument. In a different context the philosopher Bryan Magee has argued persuasively that many of our high level judgements and skills have no verbal component at all.8 Playing a violin, discriminating between wines, judging whether someone is trustworthy, admiring a painting, deciding whether two colours clash—all of these activities can occupy our full attention without being in any way verbal. Magee writes that even when one is struggling to write down one’s thoughts, one has to know what one wants to say before one chooses the words which express it best. Writers frequently revise sentences again and again, a nonsensical situation if one believes that their deepest thoughts are already verbal in form. Clearly there is more to being human than possessing language, but language has the advantage among our skills of being easy to investigate. With due apologies, I will therefore concentrate on what is known about it, while hoping that eventually scientists will move on to the consideration of our other peculiar skills. It is well known that the structure of adult human throats is substantially different from that of all other mammals, and that this enables us to produce a much wider range of sounds than, for example, chimpanzees. Like most mammals, human babies have a relatively high larynx which connects to the nasal cavity when swallowing, so that babies can breathe at the same time as suckling. The position of children’s larynxes drops by the age of seven, and

Perception and Language 19

has the unfortunate consequence of making us uniquely susceptible to choking on food. This design fault results in a significant number of deaths every year and could not exist unless there was an important compensating advantage. It is clearly a genetic adaptation enabling us to communicate by speech more efficiently. It would be strange if the changes in our vocal apparatus were our only adaptations to the use of language, and there is in fact plenty of evidence for the existence of a specific inborn language ability. There is an inherited disease, called Specific Language Impairment, which does not involve impairment, of the general intelligence. Conversely people with Williams’ syndrome, associated with a defect on chromosome 11, are very fluent conversationalists with large vocabularies, but their IQ is typically around 50. There have been a few well documented cases of children who have not had the opportunity to start learning to speak until an advanced age. If they start before the age of about six, they are usually able to catch up the missing ground and develop normal speech skills. If they start learning to speak after that age the task becomes steadily harder and the eventual skill acquired becomes progressively lower. More compelling, because of the numbers involved, are surveys of the acquisition of English by Korean and Chinese children who have immigrated into the USA at various ages. If they arrive by the age of six, then their eventual language skills are indistinguishable from those of people born in the USA; for those who arrive at a later age the eventual ability in speaking English depends upon the age of arrival.9 This is related to the existence of critical periods for the acquisition of a number of skills, and is explained in terms of neural systems degenerating or being rewired if they are not used at the ‘expected’ stage of development. Thus kittens brought up in a limited environment with no vertical lines are later unable to distinguish them: the relevant part of their visual cortex is redeployed if it is not stimulated during the critical period. The ability of human adults to discriminate sounds is strongly dependent upon their own language: many Europeans simply cannot hear the differences between different Chinese names because they are not sufficiently sensitive to pitch. Similarly the difficulty which Japanese have in distinguishing between the sounds ‘l’ and ‘r’ is based upon changes in the physical circuits in their brains; this occurs in response to the range of sounds they hear around them from a very early age. Another type of evidence for specific language skills is the astonishing rate at which words are learned in the early years. Tests of USA high school graduates show that on average they know the meaning of about 45,000 words, or up to 60,000 if one includes proper names. This implies that they have learned about nine words per day since birth, most of which are acquired without any apparent effort. Indeed in the first few years of life the rate of learning is even higher. Contrast this with children’s difficulties in learning to read. Here progress depends upon formal education programmes, which require considerable perseverance on the part of both teachers and children. Although almost nobody fails to learn to talk, the number of people who are illiterate is very substantial, in most cases because of inadequate teaching. The implication is that we have

20 Language

evolved neural circuits which make spoken language easy to acquire, but that this has not happened for writing in the six thousand years since the first written language appeared. The language instinct of humans is genetic in origin. This does not mean that there are genes for particular grammatical constructions, nor does it imply that there must be a deep ‘Universal Grammar’ as Chomsky once used to argue. Genes code for the production of proteins, and the route from proteins to specific language skills is bound to be complicated and indirect. The fact that a ‘faulty’ gene may lead to a particular failure in grammar production does not imply that the gene is responsible for that feature: the failure of a resistor may stop a TV working, but that does not mean that the resistor is more responsible for the picture than several hundred other components. There is much evidence that the use of language enables us to memorize events much more precisely, because the stimulation associated with the use of language facilitates a further spurt of brain development. There have been extended attempts to teach chimpanzees the use of language by bringing them up in human family environments. Since they do not have the vocal apparatus for speech, they have been taught using American sign language. It has proved possible to teach chimpanzees up to a few hundred words in their first five years of life, a tiny fraction of what human children achieve.10 The comparative abilities of human children and chimpanzees are rather similar until the point at which language develops in the children, somewhere between their first and second birthdays, after which our mental development accelerates away from that of chimpanzees. A related point is that we have very few memories of the period before we learn the use of language. It is obvious that our use of language does not merely enable us to communicate, but that it also profoundly affects the way we perceive the outside world. Recent experimental evidence confirms that the environment in which animals live changes the physiology of their brains. Post-mortem examinations show that rats raised in an enriched environment have thicker cerebral cortexes with more nerve fibres than other rats. Until recently it was thought that brain structure is largely fixed by adulthood, but there is now evidence that when middle-aged rats are placed in an such an environment, their brains grow substantially. The following two examples provide recent evidence for the effects of learning on the wiring of neurons in human adults. It appears from a variety of recent experiments on both humans and monkeys that certain types of repetitive strain injury suffered by typists and musicians are not caused by damage to the tendons. It appears that the abnormal use of the affected digits eventually leads to the brain rewiring the relevant circuits in a manner which prevents them working properly. The abnormal neural connections have been observed directly, and in some cases appropriate retraining can reverse the problem by causing the brain to re-rewire the neurons back to a more normal pattern. London taxi drivers are required to pass very demanding examinations relating to street layout and navigation: acquiring the necessary skills may take a few

Perception and Language 21

years. It has recently been discovered that their right posterior hippocampuses enlarge slightly but progressively the longer they do their jobs. The fact that this change is progressive demonstrates that it is related to the actual acquisition of the spatial skills. It provides good evidence that the brain retains substantial plasticity into adulthood.11 The extra stimulation we receive from the use of language almost certainly leads to the formation of extra synaptic connections in early childhood. This in effect makes us into a different animal from what we would be in the absence of such stimuli. We can easily imagine a feedback cycle operating between the development of society and of the human brain. When the adults of a tribe develop skills which aid its survival, their young learn those skills more rapidly because of their greater brain plasticity, and this makes it easier for them to develop new skills of a similar type. The size of this effect would depend on the degree to which the structure of the brain is set at birth. We know that for primates and particularly humans this is very low by comparison with other animals. As pre-human and prehistoric societies became more complex the most successful individuals might be the ones whose brains were the least fixed at birth, because they would be the most able to learn the skills which their culture required. They would survive to breed more frequently and pass on their superior ability to learn to their offspring. This would enable another round of development of the complexity of the social group. Eventually, this process leads to a genetic change in the species by purely Darwinian mechanisms. The above already shows dangers of developing into a ‘Just So’ story, and we will not pursue it further. Many hypotheses have been put forward concerning the reasons for the initial development of language, but it is difficult to test them scientifically. One idea depicts human intellectual development as the progressive growth of ever more sophisticated strategies for the purpose of deceiving and gaining advantages over neighbours. Even the date at which the human throat developed in its present form is unknown, because it is composed of soft tissues which are not preserved after death. We do know that sophisticated stone technology and cave painting existed about forty thousand years ago, when homo sapiens was already well established, but much of what is written in this field has a rather slender factual basis. What are the implication of these ideas for our mathematical abilities? It is probable that we did not need the ability to count to more than a dozen or so until the last ten thousand years. Current research indicates that the ability to distinguish numbers up to about 4 depends upon circuits which act at a preconscious level.12 It appears that formal computational arithmetic uses circuits in the brain which are also involved in generating associations between words. In contrast numerical estimation shows no dependence on language and relies primarily on visio-spatial networks of the left and right parietal lobes. Together these results suggest that human estimation skills, which are shared with animals and already present in pre-verbal infants, have a long evolutionary history. On the other hand our development of advanced mathematics could only have arisen within a culture possessing a formal system of education.

22 Language

It is evident that we do not have sense organs which enable us to perceive the meaning of large numbers such as 127928006054345 or to gaze directly at some abstract mathematical world. Our reason is not a kind of a sense organ: the knowledge which we obtain using it depends heavily upon the culture and period in which we live. People become good at mathematics for the same reason that they become good at swimming or music, by devoting their energies to developing the relevant skills over a sufficiently long period. They become truly outstanding by being obsessively interested over a period of ten years or longer. Ramanujan was just such a person. One of the most extraordinary mathematical geniuses of the twentieth century, he was born in India in 1887. As a child he displayed ability in a wide variety of subjects, but from the age of fifteen started to devote himself to mathematics to the exclusion of all other interests. By conventional standards his knowledge was extremely limited, but he developed insights into number theory which led Hardy to invite him to Cambridge, England in 1914. Before his death in 1920 from a protracted illness he had written down enough unproved new results to keep other mathematicians in work for several decades. His best parallel in more recent times may be Paul Erdos, a Hungarian who literally lived for mathematics, abandoning any semblance of a normal life as he wandered from country to country seeking problems to test his wits on. Of course mathematicians are not merely people who are good at arithmetic. There is little likelihood that we could have evolved any specifically mathematical genes over the last few thousand years, but the following facts hint at one of the possible sources of mathematical ability. Many mathematicians have particularly strong spatial imaginations, in common with architects, artists, and brain surgeons, and this might well have had advantages for hunter-gatherers travelling large distances every year. Spatial ability seems to be somewhat correlated with left-handedness, which is in other ways (increased susceptibility to auto-immune diseases and decreased life span) an evolutionary disadvantage. Left-handedness is also partially inherited and may be an example of a balanced polymorphism.13 Mathematical ability may be a result of combining the functions of the basic number module, spatial visualization skills, and general reasoning powers, reinforced by appropriate education from an early age. The extent of this ability is perhaps surprising, just as the development of a trunk in the elephant is surprising; but ultimately there are no deep philosophical conclusions to be drawn from the ability of a very small proportion of people to do advanced mathematics. It is a contingent reality. If it were not so we would no doubt devote our considerable energies to puzzling over some other issue.

Social Aspects of Language Vision provides information about the immediate environment, but the great majority of speech involves remote events or social interactions. The purpose

Perception and Language 23

of this section is to demonstrate that the understanding of even simple sentences involves enormous prior knowledge. It is also relevant to arguments against scientific reductionism, discussed in more detail in Chapter 9. Consider the following sentence, entirely typical of those which make up our everyday conversation. Joanna is happy because her daughter, Catherine, has done well in her A level examinations.

The implication that Joanna is the mother of Catherine is not as straightforward as it appears. Society has already divided motherhood into three different types, legal, genetic, and womb motherhood, and it is already possible for a child to have a different mother of each type. It may soon be possible for a child to have a womb mother and a clone father, but no genetic mother. What will have become of the concept of motherhood in another hundred years is anyone’s guess. What is certain is that Aldous Huxley’s Brave New World can no longer be regarded as science fantasy. From the two names and the reference to A levels we may reasonably guess that Joanna is British. This is not the same as saying that she has a British passport, since the passport might have been obtained by a bribe. Nor does it mean that her ancestors were British. The peculiar nature of this concept is illustrated by a shameful episode in 1968, when the British Government introduced the concept of patriality in order to reduce the number of East African Asians who could enter the UK using their British passports. Effectively the Government decided to split the concept of British nationality into two for political reasons. The concept of examination is very important in our society, but it is indeed a concept, not a physical event. British schoolchildren prepare for examinations by undergoing mock versions, which are done under more or less identical physical conditions to the true examinations. The main difference between the true and mock examinations lies in the beliefs of the pupils and others about their significance. We have seen that the simplest of sentences can combine concepts of a very abstract character. A few of these are objectively physical, but most refer to complex social institutions. Let us now look at the sentence as a whole. This might have related to a real occasion or be from a novel, but because of the context of this book you read it in quite a different way: the issue of concern was the interpretation of everyday sentences. We conclude that the significance of a sentence may be entirely altered by the context in which it was written. In fact many people believe that language evolved to facilitate social interactions rather than to communicate information about the outside world. There is good support for this in today’s world. One of the reasons why (British and probably all other) politicians are so annoying is that we, their audience, keep hoping that they might answer the questions which they are asked. They are playing a completely different game, namely using the interview or speech to persuade people to vote for them. They have achieved positions of

24 Language

power precisely because of their ability to deflect difficult questions, and to turn people’s attention to issues which will show them in a better light. Scientists (and many others) tend to think that questions should be answered honestly, and languish in obscurity because we do not have the skill to use words to such advantage.

Objects, Concepts, and Existence Although much of our daily use of language is heavily linked to our social structures, we also use it to analyse the world around us. The goal of this section is to establish that language frequently does not truly reflect reality; this problem is not capable of resolution because the number of words we can remember is far more limited than the variety of phenomena we wish to describe. For example, it is obvious that colours merge into each other continuously: there is no point in the passage from red to yellow through various shades of orange at which one can point to a physical or psychological boundary. Nevertheless we use discrete colour words. While the number of these can be increased, the boundaries between them are bound to remain artificial. Consider next an example much loved by philosophers ‘no bachelor is married’, relying on the dictionary definition of a bachelor as an unmarried man. On logical grounds this seems impeccable. The problem is that, in English at least, dictionaries do not define the meanings of words: they only summarize how they are used in the real world.14 This use changes over time. Thus none of the following accords well with the normal use of the word bachelor, in spite of the dictionary: a man living with a long term partner but not married to her; a recently widowed man; a forty year-old man who has been in prison since the age of fifteen. On the other hand a man who is permanently separated from his wife might well be called a bachelor. The phrase ‘bachelor girl’ suggests that at present the word bachelor has more to do with a life-style than with being male and unmarried. Of course this may change again. The Oxford English Dictionary has caught up with the fact that independence is now a key requirement in its definition of bachelor girls, but not for bachelor men. Even the nuances involved in the use of the word ‘girl’ are fraught with difficulties: university staff need to be careful about using it when referring to women students, even though only a small proportion would be offended. With the dangers of over-simplification in mind, we now turn to the word ‘existence’. Issues related to this word are at the core of many of the problems of philosophy.15 The most elementary type of existence is that of material objects, such as the Eiffel Tower. Many other objects are not accessible to us, simply because of the passage of time, and their past existence has to be inferred from documentary evidence. Going beyond that, I believe that my ancestor one thousand generations ago in the female line had a navel, even though I have no direct evidence for the existence of the ancestor, let alone of her navel. In this case my belief is based upon the acceptance of certain regularities in nature;

Perception and Language 25

this belief is not shared by those who consider that the world was created in 4004 bc. Existence problems are closely related to questions about truth. As soon as one admits even the remotest possibility that some everyday fact might be false, one is admitting that one does not know that it is true, but only believes that it is so. Possibly we, as finite beings, have no access to final knowledge, and have to content ourselves with the statement that certain statements have such overwhelming evidence in their support that it makes no sense to regard belief in them as provisional. There have been endless debates about the relationship between truth, belief, and evidence which we must pass over here. Let me only add that one is already taking a realist philosophical position by assuming that beliefs about the past are either true or false. I must confess to finding abstract discussions of such problems unrewarding, and prefer to consider particular examples which illustrate the difficulties which any general theory has to face. So let us discuss the nature of black holes, studied by Stephen Hawking and Roger Penrose between 1965–70. Their development of the earlier, non-relativistic theory of black holes depended upon the general theory of relativity, and led to the following conclusions. If a star is sufficiently massive (a few times the mass of the Sun) then it eventually turns into a supernova. The remnant after the supernova explosion may still be so massive that any light or other radiation which it emits is unable to escape beyond what is called its event horizon. In the theory the remnant, called a black hole, is invisible, but it may still have gravitational effects on other nearby bodies. There is steadily increasing evidence, many would say virtual certainty, that such objects do exist. A well documented example, Cygnus X-1, is the invisible component of a binary X-ray system in the constellation Cygnus; among many other candidates are V404 Cygni and Nova Scorpii 1994. In the last few years astronomers have found exciting evidence that most and perhaps all galaxies have supermassive black holes at their centres. The one at the centre of our own galaxy has just been identified as Sagittarius A*. In spite of the accumulating evidence confirming theoretical predictions about the properties of black holes, the physics of the interior of black holes is not understood. General relativity tells us that there are singularities at their centres, but the physics of space-time near the singularities may only be explicable using quantum theory. If we believe in general relativity, we can never obtain any direct or indirect evidence about what is happening inside them. So we are expected to believe in the existence of something which is in principle unknowable—almost a religious injunction, except that it is made by obviously serious scientists.16 Although rainbows look like material objects, a little reflection shows that this cannot be true. Different people standing a few metres apart might agree that they are looking at the same rainbow, but disagree about where it meets the ground. Someone who appears (to someone else) to be standing ‘in’ a rainbow would not experience any peculiar visual effects. The simple fact is that rainbows are neither material objects nor concepts: the raindrops which cause

26 Language

our ‘rainbow sensation’ do not have any intrinsic properties to distinguish them from other neighbouring raindrops. They are only distinguished in terms of their spatial relationship with both the sun and ourselves as observers. Physics explains the phenomenon perfectly, but the structure of our language does not provide an obvious category into which they fall. We turn next to concepts. Jerry Fodor17 suggests that a concept should be defined as a list of ‘features’ stored in memory, that specifies relevant properties of the things the concept applies to. Fodor’s definition does not necessitate the existence of any things to which the concepts apply, and it places concepts firmly within the realms of space and time. Thus, in spite of the fact that we believe that unicorns do not and never did exist, we have a reasonably clear idea of what they would be like. The concept is associated with definite features, and if someone uses the word without respecting those features, they would be using it incorrectly. In heraldry, art and sculpture lions and unicorns have exactly the same status; the important issue is whether the concept is clear, not whether the animals exist. I was very embarrassed many years ago to discover that there was a suburb of South-West London called Surbiton. It had been mentioned frequently in newspapers as representing a certain type of middle class suburban political attitude, and I had concluded from the over-appropriate name that it was an invention. When I discovered during a rather confusing conversation that it actually existed, I was interested to realize that I did not need to change any of my other beliefs about its characteristics! Much later I realized that my original attitude towards Surbiton had been closer to the truth than I had thought. Many people living there no doubt regarded its newspaper image as a caricature. Its existence was irrelevant: if there had been no Surbiton, newspaper columnists would have chosen some other place to represent these particular attitudes. There is a category of entities which are neither physical nor mental, but which exist as a part of our collective culture. The philosopher Karl Popper has argued that one should accept that something such a Roman law exists, because it has observable effects on the world of physical objects: people might end up in prison because it exists when they would be executed if some other legal system existed. On the other hand it is also clear that Roman law is a human construct—five thousand years ago it did not exist. In this respect justice is rather a more difficult notion. Some would say that it could not predate human society and is a biologically innate concept, while others might believe that it emanates from God and has always existed. Another example of an entity which exists by social convention is money, to be discussed in Chapter 9 in an argument against scientific reductionism. Yet another type of existence is that of skills, such as producing an axe by knapping a stone, or playing a piano. Their existence can be proved beyond challenge by the person showing that they can perform the relevant task. A person can prove that they understand the skill in conversation, but they can only prove that they possess it by demonstration. The philosopher John Lucas

Perception and Language 27

has suggested that much of mathematics depends on the development of such skills rather than on the existence of abstract theorems.18 The peculiar relationship between time and existence is provided by the game Eternity, in which 209 irregular shaped pieces are assembled in a jigsawlike fashion on a specially designed board. In 1999 the inventor of the game, Christopher Monckton, offered a prize of a million pounds to the first person who put the entire jigsaw together correctly. The game became very popular and, to Monckton’s great discomfort, the prize was claimed in September 2000 by two Cambridge mathematicians, Alex Selby and Oliver Riordan. So one can have no doubt that a solution exists: it has made two people much richer and another much poorer. On the other hand its solution presumably could not have existed before the game was invented, so its existence has to be regarded as time-dependent. If one believes that the solution came into existence as the game was invented, should one symmetrically believe that the solution will disappear if all memories of the game are one day lost to humanity? And if some historian comes across a description of the game in some library, does the solution then come back into existence immediately or only when someone rediscovers it? If it is the same solution, where was it in the intervening period? Are these real questions or are they just about how we choose to use the word ‘exist’? We leave the question at this point, because the philosophical literature on such matters is vast, and does not appear to have led to a clear conclusion.

Numbers as Social Constructs There are two extreme views about the nature of numbers, and many others in between. One, called mathematical Platonism, declares that numbers exist independently in some objective sense, and that mathematicians are engaged in uncovering the properties which they already have. The other declares that numbers are concepts of the same type as all others in our language, invented by us, and endowed with properties which we can then investigate or modify as we see fit. The issue is not about whether numbers exist, but whether they do so independently of society or as social constructions (concepts). The Platonic position seems to be supported by the following argument, discussed at length by Benacerraf and others.19 We are not permitted to use the word truth in mathematics differently from the way we use it in all other contexts. Therefore the statement that there are three primes between 45 and 60 must be true because it refers to entities which do indeed have the properties stated. We can examine these entities (the numbers between 45 and 60) one at a time, and confirm that exactly three of them are indeed primes. As with all philosophical arguments, there are counter-arguments. Statements in ordinary language are extremely varied. Thus: There are six types of outcome to a game of chess.

28 Language

is a perfectly normal sentence, whose truth is certainly not based upon examining all possible games of chess and dividing them into exactly six groups according to their outcome. If there is any reference it is to the concept of an ending. In other cases an apparently simple statement becomes steadily more obscure the longer one thinks about what exactly is being referred to. Consider the sentence: There are five vowels in the English language.

The objects being referred to here are vowels. But what exactly are vowels? Since we consider that ‘y’ is sometimes a vowel and sometimes a consonant, it follows that being a vowel depends upon context rather than shape. It has some relationship with pronunciation, but in English one cannot decide what vowel is being used from the pronunciation. Every letter may appear in many fonts and sizes, so letters are certainly not copies of material objects. Once again we are referring to abstract objects, which have changed over time and even now vary from one language to another. The above examples indicate that the possibility of making statements about abstract entities does not imply that those objects exist independently of society. According to the philosopher Karl Popper, numbers are also simply a social construction. The infinite sequence of natural numbers, 0, 1, 2, 3, 4, 5, 6, and so on, is a human invention, a product of the human mind. As such it may be said not to be autonomous, but to depend on World 2 thought processes. But now take the even numbers, or the prime numbers. These are not invented by us, but discovered or found. We discover that the sequence of natural numbers consists of even numbers and odd numbers and, whatever we may think about it, no thought process can alter this fact of World 3. The sequence of natural numbers is a result of our learning to count—that is, it is an invention within the human language. But it has its unalterable inner laws or restrictions or regularities which are the unintended consequences of the man-made sequence of natural numbers; that is, the unintended consequences of some product of the human mind.20

In What is Mathematics, Really? Reuben Hersh writes in similar terms, but with the advantage of actually knowing about mathematics from the inside. Mathematics is human. It’s part of and fits into human culture. Mathematical knowledge isn’t infallible. Like science, mathematics can advance by making mistakes, correcting and recorrecting them. (This fallibilism is brilliantly argued in Lakatos’s Proofs and Refutations.) . . . Mathematical objects are a distinct variety of social-historic objects. They’re a special part of culture. Literature, religion and banking are also special parts of culture. Each is radically different from the others.

Let me present an example which is relevant to the question of whether mathematical ideas and mathematical theorems are invented or discovered.21 Perfect numbers are defined as numbers which are equal to the sum of their

Perception and Language 29

factors, including 1 but not including the numbers themselves. So for example 6=3+2+1 is perfect. The next three perfect numbers, 28, 496, and 8128, were all known to the Greeks, and a number of interesting theorems about them are known. In a similar spirit let us call a number neat if its number of factors, including 1 but not including the number, is a factor of the number. Thus 15 has the factors 1, 3, and 5, and the number of these factors, namely 3, divides 15, so 15 is neat. On the other hand 125 has three factors, 1, 5, and 25, and 3 is not a factor of 125 so 125 is not neat. It is easy to prove that every prime number is neat and that a product of five distinct primes is neat if and only if one of those primes is 31. A product of six distinct primes cannot be neat but a product of seven can. If p is a prime then the number p n is neat if and only if n is a power of p. Other theorems about neat numbers could no doubt be proved if one were interested, and one could make conjectures about the asymptotic density of the neat numbers in the set of all numbers. Did the class (i.e. collection) of all neat numbers exist before I invented it, specifically in order to write this paragraph? I would contend not; mathematics could well do without the concept and it would probably never have been invented but for my wish to demonstrate how easy it is to produce definitions and theorems. On the other hand once I invented the class and then discovered the theorem about products of five distinct primes, its truth was not a matter of opinion. It can be tested experimentally on a computer, and proved using standard mathematical methods.22 This is entirely in accord with other contexts in which we use the word invent. When the Wright brothers invented the aeroplane, it was nevertheless an objective fact that it could fly. Nobody has invented a matter transporter of the type familiar to Star Trek fans because inventing something, as opposed to imagining it, necessitates that it works. Similarly in mathematics one cannot invent a concept if that concept is self-contradictory. An attempt to develop a theory of pentagons for which the sum of the internal angles is an odd number of degrees leads to only one theorem: no such pentagons exist. Mathematics is relatively objective in the sense that it does not often allow for varying opinions, but that by no means forces one to the conclusion that it must be about entities which pre-exist their first consideration by human beings. Whether or not a particular idea, be it primes or neat numbers, is ever invented, depends upon cultural issues and also on whether the idea is simple enough for our brains to understand. It may also be a matter of mere chance. On the other hand declaring that numbers are ‘merely’ social constructions leaves some quite serious problems to be resolved. Few people would dispute that diplodocus had four legs, long before human beings evolved the ability to count them. Some people argue from this that numbers must have existed before we invented them. One may respond that what actually existed was the diplodocus with its various parts, some of which we choose to call legs in spite of the fact that the front ones are anatomically quite different from the rear ones.

30 Language

Only after we have developed a sufficiently sophisticated language is it possible for us to formulate a sentence involving the number ‘four’. We are then correct to say ‘diplodocus had four legs’ because this provides a good match between what we see and the concepts which we have constructed. It has been put to me that if an alien civilization were found to have been counting before the human race evolved, that would prove that numbers exist independently of ourselves. While this is true, it is not terribly profound. If an alien civilization were found to have used diagrams (or bottles), this would prove that diagrams (resp. bottles) existed before we independently thought of them. The possibility that two totally independent civilizations might have some practices in common has no deep implications. It is an interesting fact that although the use of diagrams has enabled us to organize our knowledge in a way not easily achievable otherwise, nobody appears to claim that they have some deep philosophical status. Yet diagrams dominate science almost as much as do numbers. In The Origin of Species, Charles Darwin did not use any mathematics but he did include a diagram, which he discussed for several pages; this was of a schematic tree showing the evolution of species from one or a few ancestors. The task of filling in the details of this tree has dominated evolutionary studies ever since. William Smith published the first geological map of Great Britain in 1815 after many years travelling and classifying the fossils which he found embedded in rocks. This map transformed geology into a true science and set the scene for all future work in the field. Mendeleyev’s periodic table, which classifies the chemical elements into types, still appears on the walls of almost every university chemistry department. A more recent example is the use of flow charts to explain the interactions between parts of complex projects or organizations. All of these can be described using words alone, but at the cost of becoming more or less impossible to understand. We use diagrams because they present information in a manner which our type of brain can easily assimilate. It has been suggested that the situation with numbers is quite different: it is claimed to be self-evident that alien civilizations must necessarily use numbers in the same way as we do, and this proves that numbers have an existence independent of any civilization. This is evidently pure conjecture. It is amazing that people are so confident that intelligent aliens will be essentially similar to ourselves, apart from superficial differences such as having two heads, tentacles, etc. On this planet we contemplate highly organized insect colonies and know that we will never be able to communicate with them. There have been arguments about whether dolphins have equivalent intelligence to our own, or even higher intelligence which we cannot recognize because of that very fact. How much less can we assert what undiscovered alien civilizations must be like, when we do not even have any evidence that any such civilizations exist? Finally, it is claimed that the utility of mathematics in the understanding of the physical world is so striking that this proves that it cannot just be a social construction. Hersh answers this with the blunt assertion that our mathematical

Perception and Language 31

ideas in general match our world for the same reason that our lungs match Earth’s atmosphere. One should add the caveat here that in the former case one must be referring to cultural evolution, whereas in the latter biological evolution was the driving force. But the point remains that most mathematics has grown from attempts to describe properties of the external world,23 so it is not a coincidence that the two match. Indeed after more than two thousand years of development of the subject, it would be amazing if they did not. Over the next chapters we will see evidence that mathematics is not quite as powerful as people would have us believe, and that some of its power only exists ‘in principle’. In other words there are many phenomena (such as the weather) which no amount of mathematics will in fact predict in detail. Mathematics is indeed our best tool for understanding several branches of science, and it is extraordinarily good, but it will not enable us to resolve every problem we are interested in.

Notes and References [1] This image of the nature of electricity is not scientifically accurate, but it enables us to relate its properties to things we are familiar with. [2] Ronan 1978, p. 226 [3] Mollon 1992, The procedure by which the genes (certain base pair sequences on DNA) produce proteins is described somewhat more fully on pages 214–216. [4] Mollon 1997, p. 390 [5] Davidoff et al. 1999, Shepard 1997 [6] Some people find it quite hard to achieve the effect the first time. Try removing your glasses if you wear any; experiment with defocussing your eyes, and putting your head at various distances from the page, which you must view sideways. [7] Hirstein and Ramachandran 1997 [8] He was criticizing the Oxford school of linguistic philosophers. [9] Pinker 1995, p. 291 [10] Some investigators have denied that the chimpanzees actually learn even this much, in spite of appearances. [11] Maguire et al. 2000 [12] Butterworth 1999, Dehaene et al. 1998, Geary 1995 [13] This is an inherited characteristic which has both advantages and disadvantages, preventing it from becoming either extinct or universal. Corballis 1991, p. 92–96 [14] The Académie Française has tried to regulate the French language much more strictly, but with decreasing success in recent years.

32 Notes and References

[15] I agree here with the Oxford philosopher Gilbert Ryle, who argued in The Concept of Mind that the word has more than one meaning and that the failure to recognize this was at the root of the Cartesian mind-body fallacy. This book was written at the height of the Oxford passion for linguistic analysis, now largely spent. But there are indeed situations in which the vagueness of ordinary language can seriously mislead people. [16] My colleague John Taylor has pointed out that we could indeed go into a black hole to find out if the predictions of some theory about them are correct, but we could not then tell anyone outside what our conclusions were. So a more correct statement is that we could never compare the insides of two different black holes. [17] Fodor 1998 [18] Lucas 2000 [19] Benacerraf 1996 [20] Popper 1982, p. 120 [21] A similar discussion of prime numbers has been given by Yehuda Rav [1993], but it is better to avoid a topic which many people have already encountered. [22] Postscript. It is amazing how events can overtake one. In 1998 Simon Colton’s HR computer program invented almost the same concept [Colton 1999]. The only difference is that it counted all factors, not just proper factors, and so ended up with an entirely different class of ‘refactorable’ numbers. It then turned out that these numbers had already been discovered by Kennedy and Cooper without machine aid in 1990. So my neat numbers are still original! [23] The most important exceptions are number theory and group theory. But group theory was co-opted into geometry long before its relevance to particle physics became apparent.

2 Theories of the Mind

2.1

Preamble

This chapter is largely devoted to discussions of the beliefs of Plato and Descartes. Why, you may ask, do we need to spend time discussing the views of two long dead philosophers? The answer is that their systems still exert a profound influence, in spite of their obvious faults. They have become so much a part of our culture that we rarely pause to examine them. Only by doing so have we any hope of breaking free of the constraints which they impose on our thinking. What I have to say may appear negative, in the sense that I am pointing out major flaws in belief systems without proposing a detailed alternative. My defence is that it is better to acknowledge that we are not even close to an understanding of the true nature of the world, than to comfort oneself clinging to beliefs which stand no chance of being correct. Admitting this is hard, particularly for those who have devoted their lives to the search for final knowledge. We start with a discussion of Plato, because many mathematicians declare themselves to be Platonists. For most this is just the simplest way of avoiding serious thought: they subscribe to almost none of Plato’s beliefs and have worked out no neo-Platonist position. A few are more serious in their Platonism, and among these one must mention Roger Penrose and Kurt Gödel. I do not agree with anything they say (about this subject), but at least they are sufficiently honest to have formulated views with which one can argue. We will see some of the difficulties associated with their position. We then turn to Descartes’ argument that mind/soul and body/matter are entirely different types of entity. Although this has had an enormous influence on the development of science, nobody in the seventeenth century could explain how two entirely different types of entity could interact with each other, and subsequent philosophers have done no better. Of course many explanations were proposed, including the idea that God has arranged that thoughts and bodily actions would be synchronized, although there was no causal relationship between them. But nobody has devised an explanation which commands general assent. The successes of Western science have all concerned the behaviour

34 Mind-Body Dualism

of matter, and some scientists and philosophers believe that the independent existence of the mind/soul is an illusion. On the other hand the general population remain committed to mind-body dualism, which fits in well with religious belief provided one does not examine it too carefully. We consider a number of examples which show how confused the various current views are, and demonstrate how badly a new post-Cartesian approach to these problems is needed. In the final section of the chapter we will turn to the current debate about the problem of the existence of consciousness. I explain why current computers should not be regarded as conscious, and that we ourselves are conscious of only a small proportion of the activity in our brains. The fact that some of the deepest forms of processing are not conscious suggests that our thinking is not ultimately fully rational or under our control. The precise mechanisms which correspond to conscious experiences may well be found within the next few decades, but this does not necessarily mean that we will ever be able to duplicate consciousness in machines.

2.2

Mind-Body Dualism

Plato The influence of Plato as the founder of systematic philosophy has been immense, in spite of the fact that many of his arguments have been disputed or even rejected since his time. When discussing his writings, we face the problem that his views developed and even changed during his life. In the late work Parmenides he criticizes his own theory of Forms in a dialogue between Parmenides and Socrates, and it is often not clear what the conclusion of the discussion is. He even uses arguments which appear to be incompatible in a single book. The account below is therefore a selection from his views, and almost any of the statements made could be the subject of prolonged debate. Plato frequently put words into the mouths of Socrates and others, and we often cannot tell to what extent these represented his own ideas or beliefs. The real Socrates lived in Athens in the fifth century bc, and was considerably older than Plato. He wrote nothing of his own, and is mentioned by several other Greek writers, but Plato is the main source of information about him. He was a considerable public figure, who was eventually condemned to death in 399 bc, ostensibly for ‘corruption of the young’ and ‘neglect of the gods’. The actual reason was his close association with Critias and Alcibiades, who were on the ‘wrong’ side in the political ferment of that period. Plato’s story of Socrates’ refusal to accept a lesser punishment, or to attempt to escape, is probably true. He died at his own hand by drinking poison, convinced of the immortality of his soul. One of Plato’s central ideas was the theory of Forms (the Greek word eidos is also translated as Ideas or Essences). These are ideal versions of qualities possessed by material objects to a limited and imperfect extent. They are not

Theories of the Mind 35

concepts but ideal objects possessing the properties to which they refer in the most perfect manner. Thus the lines and circles which we can draw are necessarily imperfect, but are approximations to the Forms of a Line and a Circle. Similar considerations apply to Beauty, Justice, and Equality. The Forms have a real, objective existence outside our minds, and our knowledge of them is acquired partly by recollection from an earlier existence, but also by disregarding the imperfect material world and withdrawing into contemplation. The role of the philosopher is to study the Forms, which alone are worthy of his attention because only they are permanent, perfect, and ultimately real. Plato’s principal use of the theory of Forms was in discussions of ethics and politics. In the Republic he repeatedly refers to the Forms for Justice, Beauty, and Equality: Having established these principles, I shall return to our friend who denies that there is any abstract Beauty or any eternally unchanging Form of Beauty, but believes in the existence of many beautiful things, who loves visible beauty but cannot bear to be told that Beauty is really one, and Justice one, and so on,—I shall return to him and ask him, ‘Is there any of these many beautiful objects of yours which may not also seem ugly? or of your just and righteous acts that may not appear unjust and unrighteous?’. . . Those then, who are able to see visible beauty—or justice or the like—in their many manifestations, but are incapable, even with another’s help, of reaching absolute Beauty, may be said to believe but cannot be said to know what they believe.

In mathematics abstract argument led the Greeks, and later ourselves, to an enormous flowering of knowledge, so it is understandable why Plato came to regard it as the highest type of thought. However, the development of experimental science was held back for hundreds of years by the view that the direct investigation of nature was not a suitable occupation for educated people. In the mathematical and ethical contexts Plato’s theory has considerable plausibility. However, in later works Plato did not restrict the theory in this way. The following dialogue between Socrates and Glaucon in the Republic, Theory of Art reveals a much stronger claim about the scope of his theory. ‘And what about the carpenter? Didn’t you agree that what he produces is not the essential Form of Bed, the ultimate reality, but a particular bed?’ ‘I did.’ ‘If so, then what he makes is not the ultimate reality, but something which resembles that reality. And anyone who says that the products of the carpenter or any other craftsman are ultimate realities can hardly be telling the truth, can he?’ ‘No one familiar with the sort of arguments we’re using could suppose so.’ ‘So we shan’t be surprised if the bed the carpenter makes lacks the precision of reality.’ ‘No.’ ··· ‘And I suppose that God knew it, and as he wanted to be the real creator of a real Bed, and not just a carpenter making a particular bed, decided to make the ultimate reality unique.’

36 Mind-Body Dualism

Fig. 2.1 The Carpenter’s Bed

Here the process of reification is particularly clear. Plato passes from particular beds to the concept of a Bed, and then declares that there should be an ideal object corresponding to this concept. Since he is determined that this ideal object does not simply reside in our minds or our society, it must have been made by God. Plato himself was not completely happy with applying his theory of forms to material and manufactured objects, or so it appears from Parmenides, but it is not clear that he abandoned it. Figure 2.1 is one of thousands of different designs for a bed, none of which can be the one made by God, but all of which are supposed to be pale reflections of the ultimate reality. I, for one, cannot imagine what the one true Bed could possibly be like, but Plato argues above that this merely proves that I do not really know what beds are. The unconvincing passage about the Bed should be compared with Plato’s story about the cave in the Republic. Here he likens non-philosophers to men imprisoned in a cave since childhood, and tied down so that they can face away from the light, so that they only see shadows of the true Reality cast onto the wall in front of them. Perhaps Plato was thinking about the problems which we discussed in the last chapter, but if so his response to them was quite different. He advocated withdrawal from the world of the senses, whereas we resort to scientific instruments to interpret it. Another important component of Plato’s philosophy is the pre-existence of the soul, discussed at length by Socrates in Phaedo. His argument runs as follows. We recognize that two objects are more or less equal by comparing their relationship with the Form Equality. Being parts of our material body, our senses are not capable of perceiving the Form Equality, but without an appreciation of it we could not start to make sense of the world. Therefore our understanding of it must be present at birth, and must be a memory from an earlier non-material existence. The same applies to other knowledge, which, truly speaking, is recollection from this earlier existence aided by the use of the intellect (in other places in Phaedo Plato seems to suggest that during abstract thought the soul can partially separate itself from the body and enter the immortal and unvarying world of Forms). From the fact that the soul pre-exists the body, we see that it

Theories of the Mind 37

is not mortal, following which a lengthy argument leads Plato to the conclusion that it is imperishable and necessarily survives death. Plato’s negative view of the possibility of acquiring real knowledge in this world is made very clear in the following words of Socrates in Phaedo: Because, if we can know nothing purely in the body’s company, then one of two things must be true: either knowledge is nowhere to be gained, or else it is for the dead; since then, but no sooner, will the soul be alone by itself apart from the body. And therefore while we live, it would seem that we shall be closest to knowledge in this way—if we consort with the body as little as possible, and do not commune with it, except in so far as we must, but remain pure from it, until God himself shall release us.

Plato’s attempts to provide unassailable foundations for ethics and mathematics have been criticized from many different points of view, of which we can only select a few. The first is of a linguistic nature. Both English and Greek allow one to form abstract nouns from adjectives with great ease, but one should not suppose that by using this construction one must have identified an entity which exists independently of the language. On the contrary, an abstract noun corresponds to a concept, which might well not be associated with any type of object.1 We saw several examples in the last chapter, such as Roman law and the ability to play a piano, whose meaning is highly culture-dependent. A second problem concerns the uniqueness of Plato’s Forms. It is certainly clear that there is only one concept of a bed, even though the boundaries of this concept are not well-defined. However, the claim that every Form is necessarily unique leads immediately to paradoxes, as pointed out by Bertrand Russell. The Form of a Triangle is a perfect triangle, so it must have three perfect edges, which are straight lines. So it seems that even God has to make three copies of the Form of a Line in order to produce the Form of a Triangle. We will discuss this in greater depth below.

Mathematical Platonism Many mathematicians consider themselves to be mathematical Platonists in the sense described on page 27, even though they do not believe in the preexistence of the soul, and cannot explain how one might ‘see’ mathematical Forms. Among the most famous of these are Kurt Gödel and Roger Penrose, both of whom made major breakthroughs in their chosen fields. I will say something about the mathematical considerations which (in my view incorrectly) led them to embrace Platonism on page 111, but let us consider their philosophical conclusions in their own right first. We start with Gödel. He believed that numbers and even infinite sets exist in themselves, and that any statement about them must be objectively true or false whether or not we know which is the case. He also believed that mathematical entities could be directly perceived: But, despite their remoteness from sense experience, we do have something like a perception of the objects of set theory, as is seen from the fact that the

38 Mind-Body Dualism axioms force themselves upon us as being true. I don’t see any reason why we should have less confidence in this kind of perception, i.e. in mathematical intuition, than in sense perception.

This argument is undermined by the evidence, described at some length in Chapter 1, which establishes that our sense perceptions do not give us a reliable picture of the outside world. Gödel’s beliefs are regarded as bizarre by many mathematicians and philosophers, in spite of his fame. Here is a typical comment, of Chihara: Gödel’s appeal to mathematical perceptions to justify his belief in sets is strikingly similar to the appeal to mystical experiences that some philosophers have made to justify their belief in God. Mathematics begins to look like a kind of theology. It is not surprising that other approaches to the problem of existence in mathematics have been tried.2

Roger Penrose is even more explicit about his Platonism. The following is taken from The Emperor’s New Mind: When mathematicians communicate, this is made possible by each one having a direct route to truth, the consciousness of each being in a position to perceive mathematical truths directly, through this process of ‘seeing’. . . Since each can make contact with Plato’s world directly, they can more readily communicate with each other than one might have expected . . . This is very much in accordance with Plato’s own idea that (say mathematical) discovery is just a form of remembering! Indeed, I have often been struck by the similarity between just not being able to remember someone’s name, and just not being able to find the right mathematical concept. In each case, the sought-for concept is, in a sense, already present in the mind, though this is a less usual form of words in the case of an undiscovered mathematical idea.

Penrose, like Gödel, seems to regard introspection as a reliable way of gaining insights into the working of the mind. Unfortunately we have seen that twentieth century psychological research does not support this optimism. Our impression that we have a simple and direct awareness of the world and of our own thought processes are both illusions. By way of contrast Einstein rejected the idea that the nature of reality could be deduced by the application of human reason alone: At this point an enigma presents itself which in all ages has agitated enquiring minds. How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? Is human reason, then, without experience, merely by taking thought, able to fathom the properties of real things? In my opinion the answer to this question is briefly, this: as far as the propositions of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.3

He continues by contrasting the Euclidean model of reality with his own quite different theory of relativity.

Theories of the Mind 39

The most important contributor to the foundations of set theory since 1950 is probably Paul Cohen. In 1971 he came down decisively against Platonism (or Realism as he called it) in favour of a version of formalism. He summed up his conclusions about the possibility of certain knowledge in set theory as follows: I am aware that there would be few operational distinctions between my view and the Realist position. Nevertheless, I feel impelled to resist the great esthetic temptation to avoid all circumlocutions and to accept set theory as an existing reality . . . This is our fate, to live with doubts, to pursue a subject whose absoluteness we are not certain of, in short to realize that the only ‘true’ science is itself of the same mortal, perhaps empirical, nature as all other human undertakings.4

Penrose’s degree of commitment to Platonism is unusual, but a less explicit form of mathematical Platonism is quite common among mathematicians. On the other hand most mathematicians are painfully aware that their sudden flashes of insight are sometimes wrong, and that it is essential to check them carefully for consistency with other results in the field. Proving theorems frequently involves a high level of geometrical insight, but Penrose’s idea that the soughtfor concept is already present in the mind is simply wrong in many cases. If an article in a journal provides a crucial idea or technique for a theorem which you prove, it would be strange to claim that the idea of the proof was already present in your mind. It is also difficult to see what would be meant by saying that the concepts needed for the proof of Fermat’s last theorem were already in Andrew Wiles’ mind before he started trying to prove it. The fact is that theorems and/or their proofs are sometimes wrong in spite of months or years of effort on the part of their authors (e.g. Wiles’ first announcement of the proof of Fermat’s last theorem). This contradicts Penrose’s idea that mathematicians have a direct perception of the truth. Einstein’s failure to come to terms with quantum theory is another example, but in the sphere of physics rather than mathematics. The possibility of serious error also explains why mathematics journals have careful refereeing systems. If the only issue was whether the results of research papers were sufficiently important to be worth publishing, the refereeing process would be far less painful. Penrose has replied to the above criticism by stating that it misrepresents his position.5 He fully accepts that individual mathematicians frequently make errors, and goes on to say that he was concerned mainly with the ideal of what can indeed be perceived in principle by mathematical understanding and insight. He explains that he has been arguing that his ideal notion of human mathematical understanding is something beyond computation. Here, I am afraid, he begins to lose me. He uses the words ‘ideal’ and ‘in principle’ so frequently that I cannot relate his claims to anything I recognize in the activities of real mathematicians. It cannot be denied that mathematical Platonism is superficially attractive as a means of explaining our intuition about natural numbers. Unfortunately the attractiveness of an idea is no guarantee of its correctness. In Chapter 3 we will see that the theory of numbers was created by society in a series of historical

40 Mind-Body Dualism

stages, and that we only really have a direct intuition for those at the lower end of the range. Our concept of number has been extended by the introduction of zero and negative numbers, and then real and finally complex numbers. Such changes might be explained as the result of our gaining ever clearer views of the Platonic Form of Number, but they are equally easily interpreted as the result of our changing and developing the concept of number in ways which we find useful or convenient. What mathematicians cannot do is accept the demise of Plato’s philosophical system, and then continue to refer to it as if it were still valid. A key issue for Platonists is the belief that any mathematical statement is true or false before anybody has determined this. Believing this, however, is not mandatory for mathematicians. Whether or not they are Platonists, everybody agrees that if a person has a genuine proof of some statement, it is not plausible that someone else will correctly prove the opposite. Issues relating to logical consistency do not only arise in mathematics. It uses such ideas more than most other fields, but they arise everywhere. For example, it is not possible that a chess player with white pieces can force a win and that the player with black pieces can do the same. Nor is it possible that I have a sibling but my sibling does not. A genuine mathematical contradiction involving the whole numbers would show that arithmetic is inconsistent. This is indeed (just about) logically possible, but it is not worth losing sleep over. If such a contradiction were to be discovered within arithmetic, it would not be a disaster, but a wonderful opportunity to look for a better theory. The experience of three thousand years shows that any such inconsistency must be very subtle, and it would not be likely to have any consequences in ordinary life. So far I have only quoted mathematicians’ views for or against Platonism. There is also a vast philosophical literature defending and criticizing Platonism in mathematics. In Platonism and Anti-Platonism Mark Balaguer argues as follows.6 Human beings exist within space-time. If there exist mathematical objects then they exist outside space-time (this is what eternal means). Therefore if there exist mathematical objects, human beings could not attain knowledge of them. Balaguer then discusses at length each of the steps in this argument, concluding that only his own form of Full-blooded Platonism meets all the objections. Unfortunately FBP (which he is describing, not advocating) is sufficiently different from what most Platonists mean by Platonism, that it may seem that he has abandoned Platonism altogether. This impression is heightened by the fact that he finally concludes that there is no way of separating FBP from a version of anti-Platonism called fictionalism: It’s not just that we currently lack a cogent argument that settles the dispute over (the existence of) mathematical objects. It’s that we could never have such an argument . . . Now I am going to motivate the metaphysical conclusion by arguing that the sentence—there exist abstract objects; that is there are objects that exist outside of space-time (or more precisely, that do not exist in spacetime)—does not have any truth condition . . . But this is just to say that we don’t know what non-spatiotemporal existence amounts to, or what it might consist in, or what it might be like.

Theories of the Mind 41

So in the end, the issue appears to revolve around the meaning or ‘existence’ or ‘being’; if one adopts too simple a view of this concept, it may corrupt all of one’s subsequent thought processes. We conclude with a comment of Michael Dummett, which again serves to illustrate how difficult it is to resolve such problems: We do not make the objects but must accept them as we find them (this corresponds to the proof imposing itself upon us); but they were not already there for our statements to be true or false of before we carried out the investigations which brought them into being. (This is of course only intended as a picture, but its point is to break what seems to me the false dichotomy between the Platonist and the constructivist pictures which surreptitiously dominates our thinking about the philosophy of mathematics.)7

The Rotation of Triangles The fact that Platonic Forms are eternal by definition prevents human beings manipulating them within our experienced time. On the other hand mathematicians frequently moving their mental images around as they please. In this section we discuss an example which illustrates the difficulties which certain types of Platonist can encounter. Let us consider two triangles, one inside the other. The bigger triangle has edge lengths 7, 8, 9, while the smaller one has edge lengths 3, 4, 5. We consider the problem Can the smaller triangle be rotated continuously through 360◦ while staying entirely inside the bigger one?

It may be seen from figure 2.2 that the answer is not obvious.8 While this appears at first sight to be a problem in Euclidean geometry, this is only true with qualifications. Euclidean geometry as described by a formal system of axioms has no notion of time. According to Plato geometry studies the properties of eternal Forms, and he dismisses the use of operational language with disdain in the Republic. On the other hand if one reads Euclid one finds many references to the drawing of construction lines for the purpose of providing proofs; see page 102. In his construction of spirals Archimedes refers explicitly to the passage of time and uniform rotational motion. Put briefly even the Greek geometers were not Platonists. The problem described involves two triangles with different sizes and shapes. These may be idealized triangles, but they are certainly different ones. In this problem Platonists must concede that there is not a single Form of a triangle but at least two. Indeed there must be a different Form of a triangle for every possible size and shape. This is in conflict with Plato’s insistence that there is only one Form of anything, in his discussion of beds. In this problem one is forced to concede either that the Forms of triangles may move relative to each other as time passes, or that the triangles which the mathematician is imagining are not the ideal Forms, but some other triangles which only exist in his/her

42 Mind-Body Dualism

Fig. 2.2 Two Triangles

head. Plato himself struggled to find a coherent version of his theory of Forms, particularly in Parmenides, and never decided whether mathematical objects should be regarded as Forms or as a third class between Forms and material objects. Certainly he rejected the identification of Forms with thoughts. Suppose that two people are asked to solve the problem at the same time. They are sure to start with the triangles in slightly different initial positions, and to rotate them in different ways, possibly in opposite directions, completing the task after a different period of time. So they cannot be imagining the same Platonic Forms, even if we concede that Forms are capable of moving as time passes. One could imagine that the entire population of the Earth was solving this problem at the same time. All of them could have the inner triangle moving in a slightly different manner. The obvious conclusion to be drawn from this scenario is that everybody is imagining a different pair of triangles. Every individual has access to a different abstract universe, populated with triangles which are capable of being moved under his/her volition. But this bears no relationship with Platonism, which supposes that Forms of triangles are independently existing motionless, ideal objects. From words he put into the mouth of Parmenides, it is clear that the later Plato was well aware of and troubled by this dilemma. There is a way in which Platonists can try to escape from the dilemma. It depends upon declaring that the ‘true’ problem has nothing to do with time; the fact that we think of it that way is a consequence of our defective understanding of the Platonic reality. There are several ways of eliminating any direct reference to time. The most obvious is to introduce a third space variable, turning the original question into a problem in three-dimensional Euclidean geometry.

Theories of the Mind 43

Specifically it asks whether there exists a solid body which satisfies certain rather strange conditions and which also can be fitted inside a cylindrical tube. There is no doubt that any solution to the problem can be formulated in such terms. However, I consider it perverse to declare that the problem as stated, which involves moving things around on the plane, is somehow a misunderstanding of the ‘true’ problem. This feeling is reinforced by the fact that I cannot imagine solving the problem except in the original formulation, by trying to rotate the triangle in my own subjective time. A formalization eliminating time would not simplify or clarify the solution in any respect, and would serve only to satisfy the requirements of those who demand formality. Most mathematicians find no difficulties with the problem as originally posed. It is theoretically possible that a non-constructive proof could exist, but it is hard to imagine what it would be like. Anybody solving the problem does so by devising a process for turning the smaller triangle through 360◦ within the larger one, and the process itself constitutes the solution. On the other hand the problem could be proved insoluble by finding a logical barrier to the existence of such a process. Indeed this example may be typical. J. R. Lucas has said, ‘Mathematical knowledge is very largely knowledge how to do things, rather than knowledge that such and such is the case. Claims to know how to do something are vindicated by actually doing it.’9 Mathematical truth is a very slippery concept. This is not to say that it does not exist, but rather that we cannot be absolutely sure we have found it simply because we have an apparently logical proof. People make mistakes, particularly when checking a single lengthy argument repeatedly. Our knowledge of the truth of a mathematical statement depends upon making judgements based upon appropriate evidence. This evidence includes proofs of the type presented in text books, but may also involve numerical calculations, already solved special cases, geometrical pictures, consistency with one’s intuition about the field, parallels with other fields, wholly unexpected consequences which can be verified, etc. Mathematicians try to increase their knowledge, but this knowledge is based more upon the variety of independent sources of confirmation than upon logic.

Descartes and Dualism René Descartes (1596–1650) was one of the most important philosophical figures in Europe in the second millennium. His lasting reputation would be assured by his seminal improvements of algebra and its application to the solution of problems in arithmetic and geometry. However, he transformed many areas of philosophy in a number of books which became steadily more influential after his death. This book is not the place to celebrate his achievements, since our primary purpose is to focus on unresolved problems. Therefore we will only consider the part of his metaphysics which relates to the division between mind and body. This is widely considered to be less than compelling, in spite of its subsequent influence on the development of science.

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Descartes’ famous aphorism ‘cogito ergo sum’ (I think therefore I am) and the philosophical system which he built upon it have been analysed in great detail by many scholars.10 Its precise meaning was discussed at length by Descartes himself, and it appears that he did not consider it to be a logical deduction omitting the implied statement ‘everything which thinks must exist’, which would properly need to be supported by evidence. Rather he considered his thinking, his existence, and the logical connection between them all to be equally apparent to his intuition. More important is the fact that he could entertain as a logical possibility that the existence of the external world and even of his own body were illusions created by a deceitful spirit, whereas he could not do so with respect to his mind. Thus he came to the conclusion that mind (or soul) and body must be entirely different types of entity. Descartes’ task was then to construct an entire system of belief using rational argument starting from his ‘cogito’. He recognized that reliable knowledge of the nature of the external world was extremely hard to prove, and was forced to invoke God for this purpose: I had only to consider, for each of the things of which I found some idea within me whether it was or was not a perfection to possess the item in question, in order to be certain that none of the items which involved some imperfection were present in him, while all the others were indeed present in him . . . When we reflect on the idea of God which we were born with, we see . . . finally that he possesses within him everything in which we can clearly recognize some perfection that is infinite or unlimited by any imperfection.

The first and weakest component of Descartes’ argument is that non-existence is an imperfection, and hence existence must be among the attributes possessed by God. This is close to the so-called ontological argument of St. Anselm, which had already been rejected by St. Thomas Aquinas in the late thirteenth century in Summa Theologica I q2. Aquinas is quite clear that the formation of concepts has nothing to do with existence in the Platonic or any other sense: Yet granted that everyone understands that by this name God is signified something than which nothing greater can be thought, nevertheless, it does not therefore follow that he understands that what the name signifies exists actually, but only that it exists mentally. Nor can it be argued that it actually exists, unless it be admitted that there actually exists something than which nothing greater can be thought; and this precisely is not admitted by those who hold that God does not exist.

The second component of Descartes’ argument is that God, being perfect, cannot also be deceitful. Therefore if a person has a sufficiently clear perception of some aspect of the material world, he can be confident that God would not let him be entirely misled by his senses. This leads on to his study of the nature of the world and of scientific knowledge, in which he adopts a mechanistic point of view. This was much more radical in the historical context than it might seem now. He claimed that most functions of the body did not involve the intervention of the soul, including even: the retention or stamping of those ideas in the memory, the internal movement of the appetites or passions, and finally the external movements of the limbs

Theories of the Mind 45 which aptly follow both the actions and objects presented to the senses and also the passions and impressions found in the memory.

The behaviour of animals was entirely governed by such mechanistic processes, but Descartes did allow the human mind a limited role in acts to which we pay conscious attention: Since reason is a universal instrument which can be used in all kinds of situations, whereas [physical] organs need some particular disposition for each particular action, it is morally impossible for a machine to have enough different organs to make it act in all the contingencies of life in the way which our reason makes us act.

‘Moral certainty’, he later explained, means certainty beyond reasonable doubt rather than absolute proof. The scientific philosophy of Descartes is wholly materialistic. He explained scientific phenomena by creating mechanical models to show how particles of matter interact at a scale of size which we cannot perceive directly. He countered scholastic criticisms of his approach by saying that no scientific theory could possibly be established with the same degree of certainty as theorems in geometry: And if one wishes to call demonstrations only the proofs of geometers, one must say that Archimedes never demonstrated anything in mechanics, nor Vitello in optics, nor Ptolemy in astronomy, and so on; this, however, is not what is said. For one is satisfied, in these matters, if the authors—having assumed various things which are not manifestly contrary to experience—write consistently and without making logical mistakes, even if their assumptions are not exactly true . . . But as regards those who wish to say that they do not believe what I wrote, because I deduced it from a number of assumptions which I did not prove, they do not know what they are asking for, nor what they ought to ask for.

He also emphasized the need for experimentation to distinguish between different explanations of phenomena: I must also admit that the power of nature is so ample and so vast, and these principles so simple and so general, that I notice hardly any particular effect of which I do not know at once that it can be deduced from the principles in many different ways; and my greatest difficulty is usually to discover in which of these ways it depends on them. I know of no other means to discover this than by seeking further observations whose outcomes vary according to which of these ways provides the correct explanation.

I come now to the criticism of Descartes’ philosophical system. His metaphysics has many logical flaws, which are enumerated in detail in Cottingham’s anthology.11 Even if one accepts his argument for the existence of God, only God’s lack of deceit allows Descartes to be sure that his sufficiently clear beliefs guarantee correct knowledge of the material world. History shows that this is not a reliable route to knowledge. For example, possibly convinced that his own coordinate geometry was a true description of the external world, Descartes believed that he could prove by pure thought that matter must be infinitely divisible. We now accept an atomic theory of matter, and realize that one

46 Mind-Body Dualism

of his mistakes in this respect lay in assuming that the smallest fragments of matter must have the same character as gross matter. In fact subdividing atoms into smaller fragments of the same type is not just impossible but inconceivable within the conceptual framework of quantum theory. Within the pages of this book we provide many other examples of beliefs which are absolutely clear to certain groups (or at certain times) but which are equally clearly false to others. History has demonstrated time and again that Descartes’ criterion of ‘sufficient clarity’ is so demanding that one can rarely know that it has been met. We now believe that knowledge of the material world can only be gained by testing repeatedly against experimental evidence, and that this process often leads to our having to abandon our native intuition about ‘how things must be’. If he exists, God is far more subtle than Descartes imagined. Descartes’ radical separation of mind from body was extremely convenient for the development of the physical sciences, because it enabled scientists to defer indefinitely the study of minds and to declare improper any scientific reference to final causes. It came to be believed all animal motion and eventually even human behaviour were to be described in terms of the mechanical and chemical laws governing the movement of the relevant bodily parts. Even late in the twentieth century anyone who dared to diverge from this approach risked being ridiculed by ‘true’ scientists. Eventually the behaviourist B. F. Skinner took this idea to such extremes that a retreat was inevitable. Jane Goodall’s famous study of chimpanzees in Tanzania showed that refusing to accept the relevance of goals and social relationships simply prevented one understanding their behaviour. It is possible to argue that the development of such ideas in the seventeenth century was historically inevitable because of the advance of physical science, but Descartes was the one who articulated the ideas first. In spite of the enormous impact of Cartesianism on the development of physical science, many philosophers regard it as responsible for some of our worst confusions about the nature of the world. I will pursue this issue further in Chapter 9.

Dualism in Society Debates about whether humans or animals are conscious, have free will or have souls frequently lead nowhere, because those involved in the discussion do not realize that those to whom they are talking understand the terms differently. One of the most important analyses is due to David Hume. His first book, A Treatise of Human Nature, was published in 1739, and is regarded as his most important work, in spite of the fact that at the time it was almost completely ignored. He eventually rewrote a part of it as An Enquiry concerning Human Understanding in 1748, but this was hardly more successful at first. The Enquiry contained a chapter on miracles which made clear his lack of respect for religious orthodoxy, and in 1761 all of his books were placed on the Roman Catholic Index.

Theories of the Mind 47

In the Treatise Hume demonstrated the possibility of discussing the nature of the will in a non-dualist framework. One of his main goals was to show that the common notion of free will put together two quite different ideas. He used the term ‘will’ to refer to our ability to knowingly give rise to any new motion of our body, or new perception of our mind. The word ‘knowingly’ is crucial to eliminate situations in which one is compelled to act as one does, or acts in the ignorance of the consequence of one’s actions, or is afflicted by a serious mental incapacity (madness). He emphasized that our entire social system assumes that acts of the will are influenced by consequences to the person involved. Hume contrasted the above idea with the notion of liberty, which he considered to be either absurd or unintelligible. He argued that the possibility of making choices unconstrained by rational considerations or by passions, that is randomly, is not something to be valued. Indeed he regarded it as entirely destructive of all laws both divine and human. Thus while we can easily imagine choosing randomly to eat one piece of fruit rather than another, this says little about the human condition. On the other hand a person who made an important moral decision in a deliberately random manner in order to demonstrate their free will would correctly be regarded as suffering from an abnormal personality. In spite of the force of Hume’s arguments, they have had little influence on ordinary people, who still talk and think about free will in a dualistic manner and believe that the mind/soul is radically different from the body/matter. From the religious point of view this has the advantage of allowing the soul to continue in existence when the body has been completely consumed after death. But even the non-religious may well feel that their own subjective consciousness cannot be described in the same terms as the material world. The problems are that it seems to be impossible to say what precisely the soul is while preserving both its total distinction from the body and also its ability to interact with the body. In this section we explore a few of the religious approaches to this issue, all of which have serious deficiencies. Of course the same is true of non-religious approaches: if an approach without serious deficiencies had been discovered the problem would no longer be so contentious! There is an important strand of Christian thinking which rejects dualism while retaining a belief in the afterlife. In I Corinthians Ch. 15, Paul rather enigmatically supports the idea that the soul is resurrected within a new but ideal body: So also is the resurrection of the dead. It is sown in corruption; it is raised in incorruption . . . It is sown a natural body; it is raised a spiritual body. If there is a natural body, there is also a spiritual body . . . Now I say, brethren, that flesh and blood cannot inherit the Kingdom of God; neither doth corruption inherit incorruption . . . But when this corruptible shall have put on incorruption, and this mortal shall have put on immortality, then shall come to pass the saying that is written Death is swallowed up in victory.

In the Gospel of St Matthew Ch. 13, v. 36–50 it is suggested that everyone will rise from their graves on the Day of Judgement, a notion which has been much

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elaborated in religious literature and art since then. Similarly the Nicene Creed of 325 ad refers to the resurrection of the body. Unfortunately naive dualism is still alive, indeed thriving, and various cults have embraced it with disastrous results, one of the best documented being Heaven’s Gate. In March 1997 a group of 37 adults committed consensual mass suicide in a mansion in San Diego, in the stated belief that they were embarking on a transfer of their minds to an extraterrestrial spaceship which was approaching the Earth behind the comet Hale–Bopp. The cult members were regarded as ordinary, non-threatening people by those who knew them; they ran a Web page design business and also used the Web heavily to publicize their views. It is clear that they were absolutely convinced that the mass of humanity were deluded about the true nature of the world, and that they themselves were going on to a higher stage of life in androgynous extraterrestrial bodies. According to the Exit Press Release of the cult itself: The Kingdom of God, the Level above Human, is a physical world, where they inhabit physical bodies. However, those bodies are merely containers, suits of clothes—the true identity (of the individual) is the soul or mind/spirit residing in that ‘vehicle’. The body is merely a tool for that individual’s use—when it wears out, he is issued with a new one.

The beliefs of the cult were a mixture of science fiction, mysticism, and an extreme unorthodox version of Christianity. The important point here is that their final act is incomprehensible except within a dualistic philosophy in which the mind is believed to have a separate existence from the body. In Beyond Science the theoretical physicist/Anglican priest John Polkinghorne has proposed abandoning the idea of an independently existing soul but within a Christian context. He instead describes the soul as the information-bearing pattern of the body, which dissolves at death with the decay of the body. He regards it as a perfectly coherent hope that the pattern will be remembered by God and recreated by him in some new environment of his choosing in his ultimate act of resurrection. There are two difficulties with this idea. The first is that it assumes that the mind remains active and clear up to the point of death. In the case of Alzheimer’s victims, when is one supposed to take the pattern? If this is done before the onset of the disease then many valid experiences will be lost, but if it is taken at the point of death, almost no pattern will still exist. If the pattern is supposed to refer to the sum total of all life experiences, then it cannot be reincarnated in a body, because bodies have a location in time. The second problem is that the recreation of a person from their pattern (however that term is interpreted) cannot be regarded as the same person. If there is no physical continuity between the original and the copy, then the copy is just that. If the words ‘information bearing pattern’ and ‘remember’ have their normal meanings then one has to admit that God could make two or more such copies if he chose to do so; since it is not possible that both would be the original person, neither can be.

Theories of the Mind 49

What can we learn from these examples? Mind-body dualism has been rejected by almost all current psychologists and philosophers on the grounds that the idea explains nothing. On the other hand many people continue to adopt a dualistic view of the world, while being deeply disturbed that groups such as Heaven’s Gate or the Spanish Inquisition might actually act on that belief. The search for personal immortality is clearly a deep aspect of the human psyche, although no coherent accounts of how it could be the case have yet been produced. But the strengths of people’s beliefs have not often depended on rational argument, and this one does not seem likely to be abandoned soon.

2.3

Varieties of Consciousness

We have seen that both Plato and Descartes were dualists: they believed that the soul/mind could be separated from the body/matter. Plato rejected the study of imperfect matter as worthless by comparison with his Forms, and believed that mathematical ideas had a real, independent existence which the soul could appreciate directly. Descartes had trouble explaining how minds could have reliable knowledge of the material world, and had to invoke God’s help in achieving this. His philosophy of science swept all before it, but consigned the mind to an ever smaller role in the scheme of things. It now appears that many philosophers have adopted a purely materialist view in which mind is a function of or process in the brain; they regard belief in the soul as being no more than a remnant of a long outmoded system of thought. The current debate concerns not the existence of souls but the nature of consciousness, and we will concentrate on this issue henceforth. A recent survey of some current attitudes towards the mind-body problem reveals strongly expressed disagreements on almost every issue. Current positions range from the denial of the reality of consciousness (eliminative materialism) to the statement that the solution is obvious and the suggestion that the solution is straightforward but we as humans are physiologically incapable of understanding it. An amusing comment on all this was made by the philosopher John Searle: Seen in perspective, the last fifty years of the philosophy of mind, as well as cognitive science and certain branches of psychology, present a very curious spectacle. The most striking feature is how much of mainstream philosophy of mind of the past fifty years seems obviously false. I believe there is no other area of contemporary analytic philosophy where so much is said that is so implausible.12

It is tempting to define consciousness as the ability of an entity to interact with its environment. Approaching the problem this way leads one into serious problems. While it is clear that humans and dogs are conscious, we may have legitimate doubts about ants and viruses, and few would want to allow barometers even a limited degree of consciousness. Taken literally, the definition leads us to endow everything, even atoms, with a very slight degree of consciousness,

50 Varieties of Consciousness

and to measure the degree of consciousness of an entity in terms of the complexity of its interactions with the environment. We are then forced to accept that computers are conscious, their ‘environment’ consisting of their input and output devices. This definition has the merit of simplicity and definiteness, but it trivializes many important issues relating to consciousness. A key issue is the distinction between consciousness in the third person sense: what makes other people behave as they do, and consciousness in the first person sense: what is the fundamental nature of my subjective impressions? There seems to be an underlying dualism in the way we think about consciousness, just as there is for souls, with the difference that it is harder to deny the existence of subjective consciousness. The difficulties of distinguishing between the two types of consciousness is demonstrated by the existence of visual illusions. We know of their existence because we experience them subjectively. On the other hand different people experience the same illusions when show the same pictures, so they also have an objective aspect. The illusions do not exist in the pictures themselves, but are produced inside our heads. We may eventually be able to explain them physically in terms of modules in our brains and unconscious processing, but this will not remove the subjective experiences. In the remainder of this chapter we will only discuss the aspect of consciousness amenable to scientific study: third person consciousness. A variety of ideas about the nature of first person consciousness will be described in Chapter 9.

Can Computers Be Conscious? The first goal of this section is to demonstrate that current computers are not conscious under any reasonable interpretation of the word. Then we will move into more difficult territory, with the aim of clarifying the debate rather than resolving it. It is well recognized that computers can perform certain tasks such as the evaluation of extremely complicated numerical expressions vastly more rapidly and reliably than human beings. For certain types of algebraic mathematics computer software such as Mathematica and Maple can also out-perform us by a huge margin. However, mathematicians are not on the verge of becoming redundant! The same software packages are completely lost when faced with a problem such as proving that nn+1 > (n + 1)n for all n ≥ 3. The proofs of inequalities are notorious for requiring ingenuity, sometimes to an extreme degree. Nobody has yet found a means of reducing problems involving them to routine procedures which a machine could implement. I am not claiming that computers will never be able to attack such problems, but only that programs such as Mathematica and Maple are simply expert systems, provided with a set of rules by mathematicians. They are helpless in situations in which mathematicians have not been able to develop systematic procedures, even for their own use.13

Theories of the Mind 51

An example of an expert system is the computer Deep Blue, the last in a series of chess-playing computers designed by IBM over a period of years. In May 1997 it played a series of six games against the world champion Gary Kasparov, acknowledged to be the greatest (human) grandmaster of all time, and beat him by 3 21 games to 2 21 . Deep Blue’s method of playing chess was quite different from that of a human player. It examined an enormous number of possible lines of play, using a scoring system to decide which to pursue to greater depth, and eventually choosing the optimal strategy according to rules formulated by its programmers. Its chess-playing skill came partly from its ability to examine 200 million chess positions per second, and partly from the rules programmed into it about what kind of positions to aim for and avoid. Human players, on the other hand, use an intuitive method to decide which lines of play to examine, and do not consider more than a few hundred positions in detail. I am not aware that anyone involved in the design of Deep Blue ever proposed that it was conscious or engaged in genuine thought. The fact that Deep Blue could beat Kasparov is not really as important an issue as some people seem to think. Ten years before Deep Blue’s victory chessplaying programs could already beat all but a tiny fraction of the human race. Why people should feel that their own superiority is assured if one extremely unusual individual can out-perform a computer has always been a mystery to me. The real issue is whether the processes the computer uses can be classified as conscious thinking, and in this particular case the answer is surely no. It is interesting to consider how a human being does a computation in arithmetic. During the process we do not think about the meaning of what we are doing, but simply turn ourselves into automata, processing the data according to rules which we were taught as children. Our consciousness remains, not thinking about the meaning of the rules, but monitoring our successful implementation of them and checking that our attention does not wander. There are no analogous processes in a computer, since its attention cannot wander—it has literally nothing else it could be thinking about—and its level of concentration cannot vary. Some readers may remember that the first version of the Pentium chip in the mid 1990s made occasional errors in simple arithmetic, and had to be redesigned to eliminate these. One could of course point out that humans make vastly more errors when they perform such calculations, and that we are in addition far slower. However, unlike the Pentium, we are capable of retraining ourselves if such a systematic error in our method of calculation is pointed out to us. This highly publicized accident demonstrates vividly that a computer chip is performing arithmetic mindlessly. The quality of its performance depends entirely upon its designers’ care rather than on its own abilities to think through problems. On the other hand, one can compare a pocket calculator with a mobile telephone. In spite of the fact that each can do things quite beyond the capacities of the other, they are about the same size, both have keyboards, both have LED displays and both have computer chips inside. Their different capacities result from different internal organizations of their components, and the ability of the

52 Varieties of Consciousness

mobile telephone to transmit and receive messages. So it may be with human beings. The internal architecture of our brains is totally different from that of computers, and we have the advantage that enormous amounts of information flood into our brains through our sense organs constantly. In some respects we out-perform computers and in others they out-perform us. That does not prevent both computers and ourselves being finite computing machines. Whether this is, in fact, the case is another matter. The best way of determining the answer is to try to understand how we think and copy it on a suitable machine. A similar issue arises in comparing us with eagles. As far as flying is concerned they win hands (or wings) down, but when using screwdrivers we outperform them almost as dramatically. That does not imply that there is some deep difference in our cellular structure, just that it is organized differently. A moment’s thought about the differences between animals shows how important the organization of cells/genes is to the properties of the final creature. We are said to have over 98% of our genes in common with chimpanzees,14 but even this small difference has led to remarkable differences in our intellectual capacities. We should therefore keep an open mind about the possibility that radically different hardware or software could change our view about the potential capacities of computers to think in the sense we normally use this word.

Gödel and Penrose Kurt Gödel’s importance in the foundations of mathematics, discussed in Chapter 5, is so great that it compels us to listen to his comments on the differences between human thought and that of computers. Since his views on some key issues were diametrically opposed to those of Alan Turing, almost equally important in this field, we cannot however simply defer to his authority. When one looks at what he has written, much of it seems curiously out of tune with the current views of both philosophers and scientists, who often have good reasons for not understanding what he is trying to say. At the very least his views are unfashionable, but the grounds for rejecting them should be stated explicitly. Hao Wang has reported on a number of discussions with Gödel in the early 1970s, paraphrasing his views as follows: Even if the brain cannot store an infinite amount of information, the spirit may be able to. The brain is a computing machine . . . connected with a spirit. If the brain is taken as physical and as a digital computer, from quantum mechanics there are then only a finite number of states. Only by connecting it to a spirit might it work in some other way . . . The mind, in its use, is not static but constantly developing . . . Although at each stage of the mind’s development the number of its possible states is finite, there is no reason why this number should not converge to infinity in the course of development.15

Wang tried to disentangle the utterances Gödel was inclined to make into the defensible and those which are essentially mystical. The first two sentences of the quotation embrace dualistic thought in a way which is rare in other philosophical writings of the twentieth century. The meaning of the last sentence

Theories of the Mind 53

is extremely unclear. While nobody would argue with the mind being an openended learning system, it obviously cannot literally acquire an infinite amount of knowledge. The fact that a person might in principle acquire an infinite amount of knowledge if he/she were to keep on learning for an infinite length of time has no consequences in the real world. If Gödel simply means that during an actual finite life span a person might keep on learning new facts, ideas, and techniques, then his use of the word ‘infinity’ can only serve to confuse. At earlier periods in his life Gödel expressed quite different views; for example in his 1951 Gibbs lecture he stated: On the basis of what has been proved so far, it remains possible that there may exist (and even be empirically discoverable) a theorem-proving machine which in fact is equivalent to mathematical intuition, but cannot be proved to be so, nor even be proved to yield only correct theorems of finitary number theory.

Here we have Gödel accepting that human mathematical abilities may be capable of being matched by a machine. The only problem is that neither the machine nor the mathematician would then be using provably correct algorithms. In contrast to this, let me quote from an article of Penrose published in 1995: Are we, as mathematicians, really acting in accordance with an unconscious unknowable algorithm? One inference from such a proposal would be that the reasons we offer for believing our mathematical results are not the true reasons for such belief. Mathematics would depend upon some unknown calculational activity of which we were never aware. Although this is possible, it seems to me unlikely as the real explanation for mathematical conviction. We have to ask ourselves how this unconscious unknowable algorithm, of value only for doing sophisticated mathematics, could have arisen by a process of natural selection.16

My colleague Larry Landau has recently given a careful response to this highly controversial argument.17 Brains work by trying to create patterns (i.e. mental models) which match what they learn from the outside world or from introspection. This process is almost entirely unconscious and cannot be categorized as logically sound or unsound, because it is just the operation of a physical mechanism. The procedures which we use consciously when doing mathematics are entirely different from the mechanisms which control the functioning of our brains. If our conscious thought processes are occasionally or even systematically unsound, this has no implications concerning the mechanisms used by our brains to produce these thoughts. There is even an aphorism which fits this situation: do not blame the messenger for the message. One of the popular approaches to artificial intelligence uses the theory of neural networks. Scientists in this field try to model our brains by constructing machines which learn by experience. In a very narrow sense their operation is algorithmic, in that the machines are electronic computers running programs. On the other hand the machines teach themselves how to recognize individual patterns. Often they get the wrong answer, but as time passes the frequency

54 Varieties of Consciousness

of mistakes decreases. The performance of such machines is far below what we achieve, but the machines are far simpler than our brains, so this is not surprising. Such machines function in a way which Penrose above considers to be unlikely as a model for our own thinking, but many others consider the analogy very convincing. This idea about how the brain works provides an answer to Penrose’s question about how our mathematical ability could have evolved. Pattern recognition is the primary ability of our brains, and the development of this ability over millions of years is what has made us what we are. The co-option of this ability for mathematical purposes did not need any further evolution, but depended upon the social environment becoming suitable for such activities. No special algorithms for doing mathematics exist, and no guarantees of correctness of the insights obtained are available.

Discussion Over the last decade the introduction of a variety of scanning machines has led to a revolution in the understanding of consciousness. These machines allow researchers to watch the activity in different parts of people’s brains as they are set various tasks. This is one of the most exciting current areas of scientific research, but that does not mean that it is near to solving the main problems. The human brain is incredibly complicated, and ideas are in a constant state of flux, with the eventual conclusions by no means clear, even in outline. I am not the person to review progress in this highly technical field, and will only try to describe certain issues which the final theory will have to explain. Even this is a daunting task. In scientific publications consciousness almost always refers to the third person or ‘objective’ sense of the word, that is to some unknown aspect of the neural mechanisms people and animals use when they direct their attention to a matter requiring decision making. How might we decide whether an animal is conscious in this sense? The first possibility is that we observe their behaviour, and agree to call them conscious if this is sufficiently close to our own. Let us look at it in more detail. There have been vigorous arguments among biologists about whether complicated goal-directed behaviour among higher mammals is reliable evidence for their consciousness. Indeed the admission of consciousness into animal research is quite a recent phenomenon. Injury-avoidance behaviour is often based on reflexes, and it is not completely obvious that the inner sensation of pain must be attached to it. Even in our own case pain is often felt only after the limb has been moved away. Again, many birds build sophisticated nests entirely instinctively, and may or may not be conscious of what they are doing. At the other end of the animal kingdom octopuses and squid have entirely different brain anatomies from ourselves and our common ancestor probably had no brains at all. Nevertheless they are capable of learning and memorizing facts for

Theories of the Mind 55

months. If they are to be included in the realm of conscious beings, this indicates that consciousness does not depend upon a particular type of brain anatomy. Recent investigations of the behaviour of bees when choosing a new home are particularly interesting as a test of what we mean by consciousness.18 A swarm of bees often rests in a tree after leaving its original hive, while a small number of scout bees look for possible new homes. The studies show that the decision process does not depend upon any of the scouts visiting more than one site. On this return each scout signals some characteristic of a site to the swarm by a special dance. No bee has enough brain capacity to assess the relative merits of the sites, and the decision is taken by a group process which does not require any member of the swarm to be aware of the whole range of possible new sites. Nor does it require the scouts or any other individual bees to take a final decision on behalf of the swarm. To call the swarm conscious as a swarm would be rash indeed when we have no detailed understanding of consciousness even for humans, let alone for other organisms. On the other hand it seems impossible to describe the behaviour of such a swarm without referring to goals. The above paragraph uses the words ‘choosing’, ‘look for’, and ‘decision’. Perhaps we need to use such teleological language for what are actually purely instinctive responses, because this is the only way we can relate to the physical behaviour of swarms. This problem recurs throughout the biological sciences. Returning to human beings, the distinction between conscious and unconscious behaviour is a real one. When we learn to drive a car, we are initially highly conscious of every action needed. By the time we become experienced drivers the mechanical aspects of driving have moved to the periphery of our attention and it is possible to conduct a conversation at the same time as driving. Very occasionally our attention may slip entirely and we may experience the sudden shock of realizing that we have no memory of the last traffic lights which we passed. This indicates that behaviour in humans becomes conscious not because it is complex, but when it involves unfamiliar or deliberate choices. Consider next the process of breathing. Mostly it is not under our conscious control, and even if we run to catch a bus we do not make a conscious choice to increase the rate and depth of our breathing. However, it is possible for us to take over conscious control of our lungs for short periods, and there can be no doubt that when we do this something different is happening in our brain than when we breathe unconsciously. Although it is impossible for us to tell by observing animals in the field whether they are able to control their breathing consciously, it is clear from our own case that this is a real question, and not one about the use of words. A further proof that consciousness cannot be identified with behaviour comes by considering those unfortunate people who suffer total paralysis, while retaining their mental faculties. On fortunately rare occasions this happens during surgical operations, when patients are mistakenly given the usual muscle relaxants but without sufficient anaesthetics.19 The unfortunate patients are in no doubt that their induced paralysis is totally different from anaesthesia.

56 Varieties of Consciousness

We have already discussed the phenomenon of blindsight, which illustrates well the difference between consciousness and the ability to process information. There is, however, absolutely straightforward evidence that much of our thinking is unconscious and inaccessible. This is the process of remembering facts which are not near the front of one’s mind. Many readers will remember occasions on which they wanted to remember the name of someone they had not seen for several years, or to recall a word in some foreign language which they once knew. It is possible to spend several seconds, or, as you get older, even minutes, trying to remember the required word, and then for it to pop into your mind without warning. There is something going on in one’s brain, and it is quite sophisticated since it involves the meanings of words. Nevertheless we have no idea how our minds are obtaining the required information, nor where they got it from when it arrives. One cannot simply dismiss unconscious thought as referring to low level processes. Creative thinking involves unconscious processes which are capable of solving problems which our conscious minds cannot. When thinking about an intractable problem it is common for mathematicians (and others!) to spend months trying all the routine procedures, and then put the problem aside. Frequently a completely new idea comes to them suddenly, in a flash of insight similar to that which supposedly came to Archimedes in his bath. As a typical example of this process consider Hamilton’s account of his discovery/construction of quaternions20 in 1843, following fifteen years of unsuccessful attempts: On the 6th day of October, which happened to be a Monday, and Council day of the Royal Irish Academy, I was walking to attend and preside, and your mother was walking with me along the Royal Canal; and although she talked with me now and then, yet an undercurrent of thought was going on in my mind, which gave at last a result, whereof it is not too much to say that I felt at once the importance. An electric current seemed to close; and a spark flashed forth . . .

In Science and Method, 1908 Henri Poincaré wrote in almost identical terms about flashes of insight he obtained during his study of the theory of Fuchsian functions. He also mentioned that these flashes were not invariably correct, as did Jacques Hadamard in The Psychology of Invention in the Mathematical Field, 1945. In some way prolonged and unsuccessful attempts to solve a problem stimulate one’s unconscious mind to search for new and fruitful lines of attack. When something seems (to the unconscious mind) to have a high probability of leading to the solution, it forces the idea to the mathematician’s conscious attention. The criteria used by the unconscious mind are certainly not trustworthy, and the ideas so obtained have to be checked in detail. On the other hand the phenomenon often achieves results which conscious, rational thought cannot. For the above reasons, I take it as established that the distinction between conscious and unconscious thought is a matter of fact: we do not just attach the

Theories of the Mind 57

label consciousness to all sufficiently high level processes in our brains out of convention. Consciousness does not imply the ability to perform actions, and does not arise in many types of sophisticated brain activity. There is a specific brain mechanism whose activation causes conscious awareness. We do not yet know what this mechanism is or how it interacts with other parts of the brain, but it is likely that this will be elucidated over the next twenty years. We now come to a further difficulty. Human beings have a higher type of consciousness than any other animals. Only humans and the great apes can recognize that the images they see in mirrors are of themselves and that unusual features such as marks seen on their foreheads might be removed. Only in their fourth year do children start to recognize the possibility of false beliefs in themselves and others, to remember individual events for long periods and to be able to make complex plans for the future. There is an enormous research literature on the ways in which our thought processes differ from those of all other animals, and the stages at which our special abilities develop during childhood. Episodic memory, the ability to transfer individual thoughts to and subsequently from memory, is generally agreed to be of vital importance for higher level consciousness. One possibility is that higher level consciousness arises within a yet to be located physical module in the brain which deals with this ability. A person is conscious when this module is acting normally, while dreaming might correspond to a different mode of action of the module. The routine exercise of skills such as riding bicycles, walking, driving, etc. does not pass through it, but when something unexpected happens the decisions made are routed by the module into memory, from which they can then influence subsequent behaviour. Some support for the above idea may be found in the recent research literature. Eichenbaum states: The hippocampus is crucial for memory and in humans for ‘declarative memory’, our ability to record personal experiences and weave these episodic memories into our knowledge of the world around us.21

If only matters were so simple! In a recent article John Taylor pointed out the problems with all current proposals for the seat of consciousness, and eventually came down in favour of the inferior parietal lobes.22 We are evidently still far from being able to identify the hypothetical module, and it remains possible that consciousness cannot be localized in this manner. It may correspond to some characteristic wave of activity sweeping through the entire brain. Whatever the situation, suppose brain scientists succeed in identifying a neural mechanism (a module or mode of activity) which corresponds perfectly to our own subjective experience of consciousness: when it operates we are conscious and when it is disabled by brain damage or anaesthetics we are not. From the point of view of the physiologist this would be a satisfactory solution of the problem of consciousness. Suppose next that we can design a machine which has suitably rich inputs from and outputs to the external world and which contains a hardware or

58 Varieties of Consciousness

software implementation of the neural mechanisms in our own brains. Would it then be conscious in the proper subjective sense? Or would it merely be simulating consciousness? Perhaps this is incapable of being decided in an absolute sense, but we would be forced to treat such machines as conscious. The point is that the reasons for believing them to be conscious would be as good as the reasons for believing other people to be conscious. The only ‘reason’ for believing them not to be conscious would be the fact that they were built in a factory rather than grown inside someone’s womb. The development of machine consciousness may fail purely because of the difficulty of the project. The neural network approach asserts that a brain is essentially a set of neurons, and that we can duplicate that function of the brain without worrying about other aspects. This leads to two problems. The first is that there are many key aspects of brain function which are not neural, depending upon complex chemical messengers such as endorphins and neuropeptides. The second is that neurons can grow new connections in response to external stimuli and internal injuries. The way in which they ‘know’ where to grow is under genetic control in a general sense, because everything is, but it also depends heavily on other factors which we hardly understand at all. Whether it may one day be possible to produce a neural analogue of the brain depends upon the nature of its organization. A human brain has about 1011 neurons, each possessing around ten thousand synapses, far more than any electronic machine at the present time. There is no evidence that it is possible to copy the activity of the brain in some device which is radically smaller. As soon as one looks at animals whose brains are ten times smaller than ours, one sees that all of our advanced skills have disappeared. Even chimpanzees, which have brains with about one-quarter of our number of neurons, cannot begin to match our mental achievements. It is of course possible that the organization of our brains is so inefficient that one could achieve as much with a very much smaller number of efficiently arranged components. If this is the case it is worth asking why we have not evolved such a more efficient brain, since its energy requirements impose a very heavy burden upon us: although it weighs less than two kilos, about 20% of our energy is devoted to keeping it running. This figure is so large that there must have been a very large evolutionary incentive for our brains to become more efficient. The main hope of success of the research programme depends on the brain being massively redundant, so that a much smaller and simpler system may still capture its essential features. We do not know whether this is so, but the effort of finding out will surely teach us an enormous amount. The interaction between neuro-anatomists and those involved in the design of neural networks promises to bring understanding and possible treatment of mental disorder whether or not it leads to a proper model of consciousness. One should not underestimate the magnitude of the task. Our thinking appears to be controlled by intuitive judgements about whether the procedures we are adopting are appropriate to the goals which we set ourselves, rather than by logical computation. We do not understand how to design a computer program which

Theories of the Mind 59

could copy this behaviour, even with the recent advances in neural network theory. To declare that all will become clear with a few more years’ research is to make a declaration of faith rather than a sober assessment of the current position. Let us again suppose that there is a specific brain mechanism which is involved in conscious behaviour in humans. This mechanism must be linked to a large number of modules relating to motor functions and acquired skills. There is now experimental evidence that the precise forms and even locations of these modules vary from person to person, and these will influence the expression of the consciousness mechanism. As we learn more about people’s brains and how they develop, the notion of consciousness (in the third person sense) will become much more precise and complicated. Whether this will also solve the problem of subjective consciousness is a matter of debate, as we will see in Chapter 9.

Notes and References [1] In the scholastic tradition the view I am advocating is called conceptualism, and is contrasted with Plato’s realism—which I prefer to call Platonism. [2] Chihara 1990, p. 21 [3] Einstein 1982a [4] Cohen 1971 [5] See Penrose 1996, 6.2.1 This article also contains links to a number of articles by his critics, several also writing in ‘Psyche’. [6] Balaguer 1998, p. 22 [7] Dummett 1964, p. 509 [8] In fact it is not quite possible. [9] Lucas 2000, p. 366, 367 [10] Cottingham 1992 [11] Cottingham 1992 [12] Searle 1994 [13] This may change with the development of ‘genetic algorithms’, but it is too early to say how far this idea will go. [14] Recent studies suggest that this figure should not be relied on. [15] [16] [17] [18]

Wang 1995, p. 184 Penrose 1995, p. 25 Landau 1996 Seeley and Buhrmann 1999, Visscher and Camazine 1999

60 Notes and References

[19] In 1999 anaesthetists did not have a guaranteed method of measuring the depth of anaesthesia. [20] See page 71 for further details. [21] Eichenbaum 1999 [22] Taylor 2001

3 Arithmetic

Introduction Much of modern science depends on the heavy use of mathematics, justly known as the Queen of the Sciences. Let me list just a few of its many achievements. Euclid’s geometry was for two millennia the paradigm of rigorous and precise thought in all other sciences. The introduction of the Indo-Arabic system of counting and the logarithm tables of Napier and Briggs early in the seventeenth century were vital for the development of navigation, science and engineering. Newton was forced to present his law of gravitation in purely mathematical terms: he tried and failed to find a physical mechanism which would explain the inverse square law. Darwin’s epochal theory of evolution was entirely nonmathematical, but the recent development of genetics and molecular biology have been heavily dependent on the use of mathematical techniques. The two technological revolutions of the twentieth century, quantum theory and computers, have both been highly mathematical from their inception. Indeed the two pioneers in the invention of computers in the 1940s, Alan Turing and John von Neumann, were both mathematicians of truly exceptional ability. The success of mathematics in so many spheres is a great puzzle. Why is the world so amenable to being described in mathematical terms? Albert Einstein described this as the great puzzle of the universe, joking that ‘God is a mathematician’, and many distinguished scientists have echoed this sentiment. Both mathematicians and philosophers have believed at various times that our insights into Euclidean geometry, Newtonian mechanics, and set theory/logic are exempt from the general limitations on human knowledge. Unfortunately each has eventually been proven not to have any such status; we will see how this happened later on. In this chapter I explain why our concept of number is also not nearly as simple as most people consider. A long historical process has resulted in the creation of a powerful structure which we now use with confidence. But this must not blind us to the fact that the properties of numbers which we regard as self-evident were not always so. For a mathematician to cast doubt on the independent existence of numbers might seem bizarre. Read on, and I will try to persuade you that this is actually irrelevant to the pursuit of mathematics. What we actually depend upon is

62 Small Numbers

a set of rules for producing theorems, together with informal procedures for generating intuitions about those rules. It would be psychologically convenient if the rules concerned were properties of some external set of entities. Many mathematicians behave as if this were the case. But it is not necessary. The meanings of road signs are entirely conventional, but they nevertheless explain a lot about the flow of traffic. Nobody suggests as a result that green→go and red→stop are fundamental laws of nature. In both cases one can simply accept that if these are the rules, then those are the consequences. You might also ask, if some mathematicians believe that numbers do not exist independently of ourselves, why do they bother to pursue the study of their properties? The answer is the same as might be given by musicians and artists. The process of creation, and of appreciation, gives enormous pleasure to those involved in it, and if others also find it worthwhile, so much the better.

Whole Numbers For the purposes of this discussion we will divide numbers (natural numbers, positive integers) into four types according to the following rules: one to ten thousand—small ten thousand to one trillion—medium one trillion to 10100 —large much bigger than that—huge

Here a trillion is a million million and 10100 is 1 followed by 100 zeros. I do not insist on the exact boundaries between these ranges, and would accept, for example, that the small numbers might include everything up to a million. However, there are real differences between the four ranges, which I need to describe in order to set the scene for the arguments below. The above separation of numbers into different types is similar to the division of colours according to their various names: the fact that the categories overlap and that people may disagree about the borderlines does not make the distinction a worthless one. I will not plunge straight into asking what numbers really are, since this might rapidly become either a technical discussion in logic or a philosophical debate1 . Perhaps examining the history of counting systems will prove a more enlightening way into the subject.

Small Numbers These are numbers which one uses in ordinary life to count objects. For example ten thousand represents the number of points in a square of 100 × 100 points. It is also the number of steps which one takes in a walk of two to three hours. Our present notation for counting in this range functions so smoothly that we can easily forget the history behind its development.

Arithmetic 63

In Roman times the numbers 5, 50, and 500 were represented by different symbols, namely V, L, and D. Instead of having separate symbols for each number from 1 to 9, they used combinations of a smaller number of symbols. Since this system is now almost entirely confined to monuments recording births and deaths of famous people, I summarize its structure. The symbols used in what we call the Roman system are I = 1 V = 5 X = 10

L = 50

C = 100

D = 500

M = 1000.

The integers from 1 to 10 are represented by the successive expressions I

II

III

IV

V

VI

VII

VIII

IX

X

those from 10 to 100 in multiples of 10 by X

XX

XXX

XL

L

LX

LXX

LXXX

XC

C

CM

M.

and those from 100 to 1000 in multiples of 100 by C

CC CCC

CD

D

DC

DCC

DCCC

Thus the date 1485 is represented by MCDLXXXV, the fact that C is before D indicating that the C should be subtracted rather than added. The final year of the last millennium is MCMXCIX; a simpler but less systematic notation is MIM. The system described above is only one of a number of variations on a common theme. In medieval times a wide range of notations was used. One convention put a line over numbers to indicate that they were thousands, so that IVCLII would represent 4152. Another was to separate groups with different orders of magnitude by dots, so that II.DCCC.XIIII would stand for 2814. The Romans themselves used the symbol  | to represent 1000, and the right hand half of this, D, came to represent a half of a thousand, namely 500. The tomb of Galileo Galilei in Basilica di S. Croce in Florence records his year of death as CIC .IC .C.XXXXI. This is somewhat puzzling, since he died in 1642, not 1641, by our calendar. The explanation is that in Florence at that time the year started on 25 March, and Galileo died in January. This is only one of several problems one has to confront when converting dates to our present calendar; another is that historians often do not indicate whether they have carried out the conversion or not, leading to further scope for confusion! The task of multiplying two Roman-style numbers is not an easy one. Consider the following apparently very different formulae, which use the

64 Medium Numbers

seventeenth century multiplication sign. IV × IX XL × I X I V × XC XL × XC I V × CM CD × I X

= XXXV I = CCCLX = CCCLX = MMMDC = MMMDC = MMMDC

From our point of view these all reduce to 4 × 9 = 36, with zeros attached in various places. In the Greek and Roman worlds people had to learn a new set of tables for each order of magnitude; the procedures were sufficiently complicated that whole books such as Heron’s Metrica were devoted to what we now regard as routine arithmetic. An alternative was to turn to the use of abacuses, which were certainly well known in classical Greece, and may well be of Babylonian origin. Archimedes devoted his book The Sand-Reckoner to the description of a very complicated system for representing extremely large numbers. The far better Hindu-Arabic system of counting was committed to writing by Al-Khwarizmi between 780–850 ad. It is, however, certainly much older than that. It came into use gradually in Europe between 1000 ad and 1500 ad, and was eventually to sweep everything else away. Its superiority relied ultimately upon the Hindu invention of a symbol for zero. The importance of this is that one can distinguish between 56 and 506 or 5006 by the presence of the zeros, and so does not need to have different symbols for the digits depending on whether they represent ones, tens, hundreds, etc. We now take all of this for granted, but the systematic use of zero was a long drawn out process, whose impact may be as great as that of any other single idea in mathematics.

Medium Numbers The number one trillion is so big that it is not possible for one to reach it by counting. To prove this I describe a way in which one cannot get seriously rich. Suppose you could persuade someone to give you every dollar bill which you could mark a cross on. You settle down to marking one bill per second and decide to work a ten hour day. After one day you have earned $36,000 and after a working month of 25 days you have accumulated $900,000. At the end of your first year you have approximately $10 million. At this rate you would take 100,000 years to reach a trillion dollars. Even if you could speed the process up a hundred times you will still get nowhere near a trillion dollars in your lifetime. The only way to accumulate this sort of money is to emulate Bill Gates or become the dictator of a very wealthy country. There are, however, ways in which one can make the number easier to imagine. A one kilogram bag of sugar contains about one million grains, each

Arithmetic 65

about one millimetre across. To acquire one trillion grains of sugar one therefore needs a million bags, which occupy about a thousand cubic metres. This would fill all of the space in two or three of the semi-detached houses of the London street in which I live. In spite of their enormity, such numbers are important in our modern world, since several national economies have GNPs of this order. This was not always the case—when I was a boy a British billion was what we now call a trillion, and the conflict with USA usage hardly mattered because numbers of this size almost never arose. Perhaps one reason for their recent importance is that our computers can count up to these numbers even if we cannot.

Large Numbers When we turn to scientific problems, we routinely find it necessary to go far beyond the limitations of medium numbers. Examples are the number of hydrogen atoms in a kilogram and the number of neutrinos emitted by a supernova. Hindu mathematicians had words representing some very large numbers, but until the sixteenth century there was no systematic notation for writing them down. The invention of the power notation opened up the possibility of describing very large numbers in a compact notation. We write 10m to stand for the number which we would otherwise write as 1 followed by m zeros, and 3.4 × 1054 to stand for 34 followed by 53 zeros. One must not be misled by the simplicity of this notation. 1054 is not just a bit bigger than 1051 —it has three extra zeros and is a thousand times bigger! A few examples of the power of this notation is in order. One of the notable astronomical events in recent years was the observation of a supernova exploding in 1987. In truth it exploded 166,000 years ago, but for all of the time since then the light informing us of that fact has been making its way towards us. Its distance in kilometres is (3 × 105 ) × 60 × 60 × 24 × 365.25 × 166000 = 1.6 × 1018 . The size of a measured number clearly depends upon the physical units chosen, light-years or kilometres in the above example. When we talk about large numbers below, we refer to whole numbers, all of the digits of which are significant, not to measured quantities, for which only the first few digits are likely to be accurate. The distinction between the three categories of number is easy to grasp visually. One can write down a typical randomly chosen number in each of the first three categories as follows: 1528 is small, 4852060365 is medium, 56457853125600322565752345019385012884720337503463 is large.

66 What Do Large Numbers Represent?

We have a completely unambiguous way of representing large numbers, and can distinguish between any two of them. We also have ways of adding and multiplying two large numbers, by a scaled up version of the rules we are taught in our primary schools. In other words we can manipulate large numbers satisfactorily, although they do not retain the same practical relationship with counting as small numbers do.

What Do Large Numbers Represent? It is now time to discuss the relationship between counting and the natural world. I claim that large numbers are only used to measure quantities. More precisely there are no situations in the real world in which large numbers refer to counted objects. Let us start with the number of people in the world at the instant when the second millennium ended. This was about 8 billion, so it is a medium number in my system of classification. Nevertheless even this number is difficult to define precisely, let alone evaluate. People are born and die over a period which may last from a few seconds up to several hours. There is no way of specifying either of these processes sufficiently precisely for us to be able to define a moment at which they might be considered to happen. It follows that the number of people in the world at any moment has no precise value. Another example is the number of trees in a wood. Here the problem is what constitutes a tree. In addition to well-established trees several decades old, there will be saplings at all stages of growth, down to seeds which have only just started germinating. The point at which one decides whether or not to include something as a tree may affect the total by a factor of two or more. Of course one may make an arbitrary decision, such as requiring the height of a tree to be at least one metre, but this will still leave marginal cases. Even if it happens not to, there is no particular merit in using this way of defining tree-hood. With genuinely large numbers the situation is far worse. Let us think about the number of atoms in a cat. One can estimate this by weighing the cat and estimating the proportions of atoms of the different chemical elements, but this is not counting. If we insist on an exact answer and the cat had a meal a few hours ago, do we regard the meal as a part of it, or when does it become a part? The cat breathes in and out, leading to a constant flow of oxygen and carbon dioxide atoms in and out of its lungs. At what exact stage are these regarded as becoming or no longer being a part of the cat? Clearly these questions have no answers, and the number of atoms in the cat has no exact meaning. This problem cannot be avoided in any real situation involving large numbers. In an attempt to find a physical example which involves a precisely defined large number, let us consider the atom-counting problem for air sealed inside a metal box. In this case the number of atoms is not well-defined because of the process of diffusion of gas through the walls of the box. How far into the

Arithmetic 67

walls of the box should an oxygen atom diffuse before it is regarded as a part of the walls rather than a part of the gas? Of course, one can imagine a perfectly impermeable box with a definite number of atoms inside it, but this then turns into a discussion of idealized objects rather than actual ones. The conclusion from considering examples of this type is that large numbers never refer to counting procedures; they arise when one makes measurements and then infers approximate values for the numbers. The situation with huge numbers, defined below, is much worse. Scientists have no use for numbers of this next order of magnitude, which are only of abstract interest.

Addition The notion of addition is more complicated than we normally think. There are in fact two distinct concepts, which overlap to a substantial extent. Suppose we are asked to convince a sceptic that 4 + 2 = 6. We would probably say that 4 stands for four tokens | | | | as a matter of definition, that 2 stands for two tokens | | and that addition stands for putting these groups of tokens together thus creating | | | | | |, which is six tokens. A similar argument could be used to justify the sum 13 + 180 = 193, but now we would have to give rules for the interpretation of the composite symbol 13 as | | | | | | | | | | | | | with similar but very lengthy interpretations of 180 and 193 as strings or blocks of tokens. This is not, however, the way in which anyone solves such a problem. We learn tables for the sums of the numbers from 1 to 9 and also quite complicated rules for adding together composite numbers (those with more than one digit). Most people make the step between the two procedures for addition so successfully that they forget that there is a real distinction. However, when one sees the difficulties a young child has learning the rules of arithmetic, it is obvious how major a step it is. To develop the point, suppose we wish to add the number 314159265358979323846264338327950288419716939937510 to itself. The rule-based approach to addition can be applied without any trouble. (At least in principle. It might need several attempts to be sure you have the right answer.) The problem is that it is very hard to argue that this is still an abbreviation of a calculation with tokens, which cannot possibly be carried out. It is all very well to say that it can be carried out in principle, but what does this actually mean if it cannot be carried out in the real world? Edward Nelson and others have advocated the idea that the addition of very large numbers means no more than the application of certain rules. This is an ‘extreme formalist position’ in the sense that it depends upon viewing the arithmetic of very large numbers as a game played with strings of digits rather than an investigation of the properties of independently existing entities. The rules are not arbitrary: they developed out of the idea of putting tokens together in groups of ten and

68 Multiplication

then groups of a hundred, and so on. But eventually the system of rules took over until for large enough numbers that is all that is left. The tokens have disappeared from the scene, since we cannot imagine huge numbers of them with any precision. To summarize, when adding small numbers we can use tokens or rules, and observe that the two procedures always give the same answer. This fact is not surprising because the rules were selected on the basis of having this property. However for large numbers one can only use the rules. In the shift from small to large numbers a subtle shift of meaning has occurred, so that for large numbers the only way of testing a claimed addition is to repeat the use of the rules. The rules are exactly what computers use to manipulate numbers. We like to feel that we are superior to them because we understand what the manipulations really mean, but our sense of superiority consists in being able to check that the two methods of addition are consistent for small numbers.

Multiplication In Shadows of the Mind Roger Penrose claimed that we can see that 79797000222 × 50000123555 = 50000123555 × 79797000222 without performing the two multiplications, as follows. Each side of the equation represents the number of dots in a rectangle whose sides have the appropriate lengths. Since the one rectangle is obtained by rotating the other through 90◦ they must contain the same number of points. Penrose states that we merely need to ‘blur’ in our minds the actual numbers of rows and columns that are being used, and the equality becomes obvious.

Notice that the matter only becomes ‘obvious’ by blurring the numbers. This is necessary since the numbers are so large that they cannot be represented by rows of dots in any real sense. One can argue that the process involved is not one of perception but one of analogy with examples such as 6 × 8 = 8 × 6, for which Penrose’s argument is indeed justified. The analogy depends upon the belief that 6 and 79797000222 are the same type of entity, when historically the latter was obtained by a long process of abstraction from the former. The fact that the product of two numbers does not depend upon the order in which they are multiplied is called the commutativity identity. Symbolically it is the statement that x×y =y×x for all numbers x and y. Its truth is clear provided the numbers are small enough for us to be able to draw the rectangles of dots. The question then becomes whether we can extend the notion of multiplication to much larger numbers in such a way that the identity remains valid. It may be shown that this is achieved by using the familiar rules for multiplication for large numbers.

Arithmetic 69

Fig. 3.1 Multiplication Using Rectangles

It may also be proved using Peano’s postulates for huge numbers (discussed below). Having found an extension of the notion of multiplication which retains its most desirable properties, eventually we decide that the extension defines what is meant by multiplication, and forget the origins of the subject. There is another reason for doubting Penrose’s explanation of why we believe the commutativity law. In order to explain this I need to refer to entities called complex numbers. In the sixteenth century Cardan and Viète showed that certain calculations in arithmetic could √ be carried out more easily by introducing nonsensical expressions such as −5, ignoring the fact that negative numbers do not have square roots. It was repeatedly observed that if one used square roots of negative numbers in the middle of a calculation but the final answer did not involve them, then the answer was always correct! It was later realized that all of the paradoxes of this subject could be reduced to justifying the use of the imaginary number √ i = −1. In 1770 Euler wrote in Algebra: Since all possible numbers that can be imagined are either greater than or less than or equal to zero, it is evident that the roots of negative numbers cannot be counted among all possible numbers. So we are obliged to say that there are impossible numbers. Hence we have had to come to terms with such numbers, that are impossible by their very nature and which, by habit, we call imaginary because they only exist in the imagination.

Clearly he did not subscribe to the belief that complex numbers existed in some Platonic realm. The first stage in the demystification of complex numbers was taken by Argand and Gauss around 1800 when they chose to represent the complex number x + iy (already assumed to exist in some sense) by the point in the plane with horizontal and vertical coordinates (x, y). The paradoxical square root of minus one was then represented by the point with coordinates (0,1).

70 Multiplication

1 + 3i

3+i

i

−1

1 −i

Fig. 3.2 The Complex Number Plane

In 1833 Hamilton approached complex numbers the other way around. He defined a complex number to be a point on the plane, and then defined the addition and multiplication of such points by certain algebraic formulae. Following this he was able to prove that addition and multiplication had all of the properties we normally expect of them with the additional feature that i = (0, 1) satisfies i 2 = −1. So within this context −1 does indeed have a square root! Hamilton’s work was revolutionary because it forced mathematicians to come to terms with the fact that truth and meaning depend on the context. Within the context of ordinary (real) numbers −1 does not have a square root, but if the meaning of the word number is extended appropriately it may do so. The same trick is used in ordinary speech. Throughout human history it was agreed that humans would like to fly but could not. Now we talk about flying from country to country as if this is perfectly normal. Of course we have not changed, but we have redefined the word ‘fly’ so that it can include sitting inside an elaborately constructed metal box. As a result of extending the meaning of the word something impossible becomes possible. It is extremely hard for us to put ourselves in the frame of mind of Euler and others, who could not believe in complex numbers but could not abandon them either, because of their extraordinary usefulness. Psychologically the acceptance of complex numbers came when mathematicians saw that they could construct complex numbers using ideas about which they were already confident. This was a revolution in mathematics, which involved abandoning

Arithmetic 71

the long-standing belief that mathematics was the science of magnitude and quantity.2 It opened up the possibility of changing or extending the meaning of other terms used in mathematics, and of creating new fields of study simply by declaring what the primary objects were and how they were to be manipulated. In this respect mathematics is now a game played according to formal rules, just like chess or bridge. The system of complex numbers is enormously useful, and mathematicians now feel as comfortable with them as they do with ordinary numbers. The important point for us is that the multiplication of complex numbers is commutative. The only reason for the truth of the commutativity identity z × w = w × z for complex numbers is that one can evaluate both sides of the equation using the definition of multiplication and see that it is indeed true. Hamilton’s subsequent invention of quaternions in 1843 was an even more revolutionary idea. These were also an extension of the concept of number, but in this case allowing the possibility that z × w  = w × z. The technicalities need not concern us, the crucial point being that nobody previously had thought that the commutativity of multiplication was among the things a mathematician might consider giving up. Hamilton’s conceptual breakthrough led to Cayley’s systematic development of matrix theory in 1858 and Clifford’s introduction of his Clifford algebras in 1878; in both of these the commutativity of multiplication was abandoned. The importance of these ideas can hardly be exaggerated. If one had to identify the two most important topics in a mathematics degree programme, they would have to be calculus and matrix theory. Noncommutative multiplication underlies the whole of quantum theory and is at the core of some of the most exciting current research in both mathematics and physics. We conclude: the fact that multiplication of ordinary and complex numbers are commutative has to be proved, and this is possible as soon as one has written down precise definitions of the two types of number and of multiplication. The fact that the multiplication of quaternions or matrices is not commutative is equally a matter of proof. Reference to the rotation of rectangles may be used to persuade children that the commutativity property is true for small numbers, but it does not provide a proper (non-computational) proof for all numbers.

Inaccessible and Huge Numbers There are known to be infinitely many numbers. The argument for this is simple—there cannot be a biggest number, since if one reached it by counting, then by continuing to count one finds larger numbers. For numbers radically bigger than 10100 , however, we have no systematic method of doing computations. By radically bigger I do not mean 101000 , even though it is certainly vastly bigger than 10100 . Nor do I mean 213,466,917 − 1

72 Inaccessible and Huge Numbers

which has just established a new record as the largest known prime. This number has just over four million digits, and would fill a very large book if printed out in the usual decimal notation. Incidentally the proof that this number is indeed a prime took 30,000 years of computer processing time. 100 By huge I mean a number such as 1010 , which has 10100 digits. The task of writing down the digits of a typical ‘randomly chosen’ number with 10100 digits in the usual arabic notation is beyond the capacity of any conceivable computer—it would take all of the atoms in the Universe even if one could store a trillion digits on each atom. Nor can one carry out arithmetic with such numbers: the problem 88

888

999

+ 99

=?

just sits there mocking our impotence. A lot is known about prime numbers both theoretically and computationally. To illustrate the latter aspect, there are exactly 9592 primes with five or fewer digits, the smallest being 2 and the largest being 99991. In order to illustrate the failure of systematic computation for huge numbers, let us define P to be the number of primes which have fewer than a trillion digits. Quite a lot is known about the distribution of prime numbers, and the prime number theorem allows one to evaluate P quite accurately. On the other hand everything we know about prime numbers suggests that the question Is P an even number? will never be answered. A Platonist would regard this as having an entirely straightforward meaning, but this does not help him or her one iota to determine the answer. In fact a Platonist is no more likely to solve this problem than a mathematician who regards proving things about huge numbers as a formal game. I frequently hear mathematicians saying that questions such as the above pose no problem ‘in principle’. This phrase makes me quite angry. It might mean ‘I know it is not actually possible but would like to close my mind to this fact and pretend that I could do it if I really wanted to’. Another possible meaning is ‘I do not regard the difficulty of carrying out a task as an interesting issue’. Either interpretation leaves the speaker cut off from the mainstream of human activities. The second amounts to a rejection of all matters associated with numerical computation, a subject which contains many challenging and fascinating problems. Even if one has a formula for a particular huge number one may not be able to answer completely elementary questions about it. Consider the famous Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . .

Arithmetic 73

The rule for generating this sequence is that each term is the sum of the previous two. Letting fn denote the nth term one may compute f100 = 354224848179261915075 f1000 = 434665576 . . . 6849228875 f10000 = 336447648 . . . 947336875. The number f1000 has 209 digits while f10000 has 2090 digits! Now let us now 100 consider fn for n = 1010 . From a naive point of view there appears to be no difficulty in knowing what we mean by this number: one just keeps on adding for an extremely long time, using an amount of paper which steadily increases as the numbers get bigger. Unfortunately in practice there appears to be no way of determining even the first digit of this number. The exact definition and the known analytic formula for fn are equally powerless to help us, because the numbers involved have so many digits. 100 Some numbers, such as 1010 , are perfectly simple to write down in spite of being huge. The following argument shows that most are completely inaccessible. One may classify the complexity of a number in terms of its shortest description. Thus 314159265358979323846264338327950288419716939937510 is lengthy when written down as above, but has a much simpler description as the integer part of π × 1050 . Defining the complexity of a number in terms of its shortest possible description is fraught with problems if expressed so briefly, because of phrases such as The smallest number whose definition requires at least a million symbols.

Such a number cannot exist, since the above phrase defines it in 73 symbols including spaces. The accepted way out of this self-reference paradox is to replace it by the phrase The smallest number whose definition using the programming language X requires at least a million symbols.

Here X could be Java, C ++ , some extension of these which permits strings of digits of arbitrary length to represent numbers, or any other high level programming language. The number defined depends on the programming language used, but for our purposes the important issue is that one does not get trapped by the illogicalities of natural language. Using this definition of complexity we 100 see that 1010 is very simple, since it requires only 13 symbols when written in the form 10ˆ{10ˆ{100}}, which C ++ is able to understand.

74 Inaccessible and Huge Numbers

Some truly enormous numbers can be written down within the above constraints. For example we can put a=9 b = 9a c = 9b d = 9c without beginning to approach the self-imposed constraints on size. Standard high level programming languages allow one to go far beyond this by means of the definition n:=1; for r from 1 to 100 do n:=nn + 1; end; N:=n; For those who do not feel at ease with computer programs, it describes a procedure for generating numbers starting from 2. The next number is 22 + 1 = 5. The third is 55 + 1 = 3126. The fourth number in the list, namely 31263126 + 1, is still just small enough to be calculated by current PCs: it has 10,926 digits and can be printed out on about ten pages of A4 paper. The fifth number is too large for any computer constructible in this universe to evaluate (in the usual decimal notation). Only an insignificant fraction of the digits in the answer could be stored even if one allocated a trillion digits to every atom in the universe. The hundredth number in the list, which we call N , is mind-bogglingly big, and little else can probably be said about it. Writing down the shortest description of a number is quite different from representing it by a string of digits. In spite of this, the following technical argument shows that using shortest descriptions does not materially alter how many numbers can be written down explicitly. We consider a programming language which uses a hundred types of symbol, including letters, numbers, punctuation marks, spaces, and line breaks. Suppose we consider numbers whose definition can be given by a program involving no more than a thousand such symbols. The total number of ‘programs’ which we can write down by just putting down symbols in an arbitrary order is vast, but it can be evaluated by using a coding procedure. We first list the hundred symbols in some order, putting the numbers from 01 to 99 and finally 00 underneath them. The list might start with: q 01 p 10

w 02 a 11

e 03 s 12

r 04 d 13

t 05 f 14

y u i o 06 07 08 09 g h j k 15 16 17 18

Arithmetic 75

Then we replace each symbol in the program with the number underneath it. So ‘queasy’ would be replaced by 010703111206. The result is to replace every program by a number with at most 2000 digits, so the total number of such programs is 102000 . Actually almost all of them are gibberish, so the number of grammatical, or meaningful, programs is very much smaller. Programs do indeed allow one to write down a few numbers which are ridiculously large, but they do not provide a systematic way of writing down all such numbers. The conclusion is that whether we define numbers by strings of digits, or by descriptions in a chosen programming language, there is little difference in how many numbers we can effectively express. Neither approach overcomes the basic information theoretic problem that there are limits on how many different numbers we can hope to write down explicitly, and therefore to what we can actually compute.

Peano’s Postulates In everyday life we constantly rely upon the idea that if two events have regularly been associated, they will continue to be so. Thus we ‘know’ that if we bring our hands together sharply, we will hear a clapping noise. We believe this not because we know anything about the physics of sound production, but simply on the basis of experience. In Chapter 9 I will discuss Hume’s criticisms of this type of induction, but here I wish to consider what is called mathematical induction. This is not the very dangerous idea that you are justified in believing a statement such as For every positive number n the expression n2 + n + 41 is prime. simply by testing it for more and more values of n until its repeated validity persuades you of its general truth. The first few values of the above ‘Euler polynomial’ are 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, . . . which are certainly all prime numbers. After testing several more terms one might easily come to the conclusion that the claim is true. Actually it is false, the smallest value of n for which the expression is not prime being n = 40. In order to prove statements about all numbers, of whatever size, mathematicians use abstract arguments based on Peano’s postulates for the integers. Peano wrote down his postulates (or axioms) in 1889. They are 0 is a number. For every number n there is a next number, which we call its successor. No two numbers have the same successor. 0 is not the successor of any number. If a statement is true for 0 and, whenever it is true for n it is always also true for the successor of n, then it is true for all numbers.

76 Peano’s Postulates

The critical axiom above is the last, called the principle of mathematical induction. It is usually written in the more technical and compressed form If P(0), and P(n) implies P(n + 1), then P(n) for all n. Here P(n) stands for a proposition (statement) such as (n + 1)2 = n2 + 2n + 1 or Either n is even or n + 1 is even. The principle of induction is not a recipe which solves all problems about numbers effortlessly, but it is the first thing to try. A present-day Platonist might say that the truth of Peano’s postulates can be seen by direct intuition. In Science and Hypothesis, 1902 Henri Poincaré adopted the Kantian view that ‘mathematical induction is imposed on us, because it is only an affirmation of a property of the mind itself’. These arguments were not accepted by Bertrand Russell and other logicians early in the twentieth century who tried to construct the theory of numbers from the more certain and fundamental ideas of set theory. If one looks at the historical record, Russell’s caution about the ‘obviousness’ of induction is certainly justified. The Greeks did not use it, although it is possible to detect hints of such ideas in a few isolated texts.3 Its first explicit use in mathematical proofs is often ascribed to Maurolico in the sixteenth century. Even today, many mathematics students who have been taught the principle are reluctant to use it, preferring to rely upon direct algebraic proofs of identities if they can. Alternatively we may regard Peano’s postulates as a system of axioms like any other. That is they are a list of rules from which interesting theorems may be proved. These theorems agree with what we know for small and medium numbers because we can see that those satisfy the stated postulates—except that the obviousness of the last one becomes less clear as the numbers increase, and lose their obvious connection with counting. If the historical record is a guide, Peano’s axioms are less obvious than those of Euclidean geometry. They may be seen as a part of a formal system of arithmetic. For an entertaining account of the complexities involved in setting up such a formal system see Gödel, Escher, Bach: An Eternal Golden Braid by Douglas Hofstadter. Paul Bernays considered that ‘elementary intuition’ faded out as numbers become larger: Arithmetic, which forms the large frame in which the geometrical and physical disciplines are incorporated, does not simply consist in the elementary intuitive treatment of the numbers, but rather it has itself the character of a theory in that it takes as a basis the idea of the totality of numbers as a system of things as well as of the idea of totality of the sets of numbers. This systematic arithmetic fulfils its task in the best way possible, and there is no reason to object to its procedure, as long as we are clear about the fact that we do not

Arithmetic 77 take the point of view of elementary intuitiveness but that of thought formation, that is, that point of view Hilbert calls the axiomatic point of view . . . However, the problem of the infinite returns. For by taking a thought formation as the point of departure for arithmetic we have introduced something problematic. An intellectual approach, however plausible and natural from the systematic point of view, still does not contain in itself the guarantee of its consistent realizability. By grasping the idea of the infinite totality of numbers and the sets of numbers, it is still not out of the question that this idea could lead to a contradiction in its consequences. Thus it remains to investigate the question of freedom of contradiction, of the ‘consistency’ of the system of arithmetic.4

Even if we adopt the first position (naive realism), we have to admit that for huge numbers, Peano’s postulates provide the only route to our knowledge of them—the only way of convincing a sceptic that a claim about huge numbers is true makes use of Peano’s postulates or something which follow from them. The following example illustrates the issues involved. The statement that 8 is a factor of 9n − 1 means that if one divides 9n − 1 by 8 then there is no remainder, or equivalently that 9n − 1 = 8 × s for some number s. A geometrical proof of this statement for n = 2 can be extracted from Figure 3.3. A geometric proof is also possible for n = 3 by decomposing a 9 × 9 × 9 cube in a similar fashion. For n = 4 one may check that 94 − 1 = 8 × (1 + 9 + 92 + 93 )

8

1

8×8

8

Fig. 3.3 Decomposition of a 9 × 9 Square

78 Infinity

by explicitly evaluating both sides. This correctly suggests the general formula 9n − 1 = 8 × (1 + 9 + 92 + · · · + 9n−1 ) for all numbers n. However, this expression cannot be checked directly for n = 10100 because the number of additions involved would take impossibly long. Also the formula involves the mysterious . . . which invites one to imagine doing something, and should not be a part of rigorous mathematics. A more formal expression would be 9n − 1 = 8 × 

n−1


9r

r=0

in which the summation symbol is given a formal meaning by means of the Principle of Induction. So eventually one has to believe that the use of this Principle is permissible in order to prove that 8 is a factor of 9n − 1 for all n. One now comes to the philosophical divide. A Platonist believes that the Principle of Induction is a true statement about independently existing objects. The alternative view is that mathematicians are investigating the properties of systems which we ourselves construct, what Bernays called thought formations. Motivated by our intuition of small numbers, we decide to include the Principle of Induction among the rules which we use to prove theorems. Theorems correctly proved within such a system are true, because truth is always understood as relative to some agreement about the context. In this view mathematics is not like exploring a country which existed long before the explorer was born. It is more like building a city, with its unlimited potential for muddle, error, and growth. We lay the foundations of each building as well as we can, but accept the possibility of collapse. If a building does fall down, we rebuild it, learning from our errors. We also examine other buildings to see if they have the same design faults. Gradually the city becomes more impressive and better adapted to our needs, but it always remains our creation. There still remains something to be said about Peano’s Principle. If one is not willing to declare that its truth is self-evident, how can one justify its use for large numbers, the ones which scientists have a real use for? In linguistic terms, if we define large numbers by the strings of digits used to manipulate them (their syntax), then we have removed the only reason for believing Peano’s postulate, which is based on the meaning of number (their semantics). This seems a fatal blow to formalists who would argue that large numbers are no more than long strings of digits. Fortunately one can prove Peano’s Principle for large number strings in a few lines using only conventional logic. This resolves the objection.5

Infinity If one can entertain doubts about the meaning of very large finite numbers, then it seems that we have no hope of understanding the infinite. My intention here is

Arithmetic 79

to persuade you that there are many different meanings to ‘infinity’, written as ∞, all of which are of value in the appropriate context. Each of them captures some aspect of our intuitive ideas about the infinite, which, as finite beings, we cannot perceive directly. In Chapter 5 we will discuss whether infinite objects actually ‘exist’, and what this might mean. The obvious way of dealing with infinity is to write down rules for manipulating it, such as ∞ × ∞ = ∞ and 1 + ∞ = ∞ and 1/∞ = 0. One quickly finds that great caution is needed in manipulating such algebraic expressions. Otherwise one may obtain nonsensical results such as 0 = ∞ − ∞ = (1 + ∞) − ∞ = 1 + (∞ − ∞) = 1 + 0 = 1 or

2×∞ ∞ = = 2. ∞ ∞ Nevertheless infinity is used in this manner by all analysts, who learn to avoid the pitfalls involved. There is a different way of introducing infinity, which is quite close to the modes of thought in the subject called non-standard analysis. Namely one introduces a symbol ∞ and agrees to manipulate expressions involving it according to the usual rules of algebra. In this context ∞ × ∞ is not merely different from ∞, but vastly (indeed infinitely) bigger. Similarly 1/∞ is not equal to 0 but it is an unmeasurably small number, called an infinitesimal. Finally ∞ is bigger than every positive integer, but ∞ + 1 is bigger than ∞. It can be shown that if one follows certain simple rules there is no inconsistency in this system, which does capture some of the properties which we think infinity should have. Note, however, that the two notions of infinity above are different and incompatible with each other. Which we decide to use depends upon what we want to do. The symbol ∞ also appears as a shorthand for statements which avoid its use. Thus writing that something converges to 0 as n tends to infinity is just another way of writing that it gets smaller and smaller without ceasing. The corresponding formal expression 1=

lim an = 0

n→∞

means neither more nor less than ∀ε > 0.∃Nε .n ≥ Nε → |an | ≤ ε. One need not understand either of these formulae to see that the infinity in the first has miraculously disappeared in the second, being replaced by the logical symbols ∃, ∀, →. The credit for providing this rigorous ‘infinity free’ definition of limit goes to Cauchy in Cours d’Analyse, published in 1821. The symbol ∞ is considered to have no meaning in isolation from the context in which it appears. Analysts agree that this type of use of the symbol does not involve any commitment to the existence of infinity itself.

80 Discussion

The above notions of infinity provide more precise versions of previously rather vague intuitions. Since there are several intuitions one ends up with several different infinities. The above is typical of how mathematicians think: we start from vague pictures or ideas, about infinity in this case, which we try to encapsulate by rules, and then we discover that those rules persuade us to modify our mental images. We engage in a dialogue between our mental images and our ability to justify them via equations. As we understand what we are investigating more clearly, the pictures become sharper and the equations more elaborate. Only at the end of the process does anything like a formal set of axioms followed by logical proofs appear. Eventually we come to behave as if the ideas which we have reached after much struggle already existed before we formulated them. Perhaps later generations do not realize that other ideas were pursued and abandoned, not because they were wrong but because they were less fruitful.

Discussion The division of numbers into small, medium, large, and huge was a device used to focus attention on the fact that successive stages of generalization involve losses as well as gains. At one extreme numbers really do refer to counting, but at the other the relationship with counting only exists in our imagination.6 The most abstract, and recent, concept of number depends upon formal rules of logic and Peano’s property. By distinguishing between these four different types of number I seem to be violating the principle of Ockham’s razor: non sunt multiplicanda entia praeter necessitatem

i.e. entities are not to be multiplied beyond necessity. The following are some positive reasons for distinguishing between the types of number. The fact that computers can manipulate large numbers with great efficiency, but are pretty hopeless beyond that, suggests fairly strongly that huge numbers are genuinely different from large ones. In algorithmic mathematics the size of the numbers involved in a procedure is one of the primary issues, and the appearance of huge numbers indicates that the procedure is not of practical use. Abstract existence proofs often provide little information about the properties of the entity proved to exist; often they do little more than motivate one to seek to find more direct computational methods of approach which provide more information about the solutions. A version of the following paradox was known as the ‘Sorites’ in the Hellenistic period, but in the form below it is due to Wang.7 It states The number 1 is small. If n is small then n + 1 is small. Therefore, by induction, all numbers are small.

It is as clear to philosophers as to others that the conclusion of this argument is incorrect, so the only issue can be to explain where the error lies. You may ask why anyone should bother about such a trivial issue. The answer is that there

Arithmetic 81

may be other arguments which are incorrect for the same reason, even though in these other cases it may not be at all obvious that an error has been made. The same applies to the exhaustive enquiries held after a crash of a commercial airliner. They cannot save the life of anyone who died, but, if the reason for the crash is discovered, it may be possible to prevent it happening again. Michael Dummett has discussed this paradox at length and raised doubts about whether one can apply the normal laws of logic to vague properties such as smallness.8 There are in fact (at least) two ways of resolving such problems, both of which would be perfectly acceptable to any mathematician, if not to philosophers. The simplest is to simply declare numbers less than a million (say) to be small and others to be big. Dummett mentions this possibility but declares it to be a priori absurd; he ignores the fact that this is precisely the way in which the law distinguishes between children and adults, another vague issue. An alternative is to attach an index s(n) of smallness to every number, by a formula such as 106 . s(n) = n + 106 Using this formula, numbers which we think of as small get a smallness index close to 1, while very large numbers get an index close to 0. The particular formula above assigns the smallness index 0 · 5 to the number one million, so if one uses this formula one would regard a million as being intermediate between small and large. We could then say that the common notion of smallness merely attaches the adjective to all numbers for which the speaker considers the index to be close enough to 1, but all precise discussions should use the index. Either of these proposals immediately dissolve the paradox. There is even a mathematical discipline which studies concepts which do not have precise borderlines, called fuzzy set theory. The status of the Peano property is different for numbers of each size. For ‘counting’ numbers its truth is simply a matter of observation. For numbers defined as strings of digits I have shown how to prove it in a recent publication. For huge or formal numbers it is an abstract axiom. Each of the three ways of looking at numbers has its own interest, and one learns valuable lessons by finding out which problems can be solved within each of the systems. We should distinguish between the features of the external world which lead one to some idea (counting in the present context), and the mathematical system we have invented to extend that initial idea (formal arithmetic). Historically it is clear that our present idea of number is far removed from that of our ancestors, and that we may never have a good way of handling most huge numbers. The belief that individual numbers exist as objects independent of ourselves is far from being accepted by philosophers. Paul Benacerraf has examined in detail a number of different ideas about what numbers might be if they do exist, coming to the conclusion: Therefore numbers are not objects at all, because in giving the properties (that is, necessary and sufficient) of numbers you merely characterize an abstract structure—and the distinction lies in the fact that the ‘elements’ of the structure

82 Discussion have no properties other than those relating them to other ‘elements’ of the same structure. . . . Arithmetic is therefore the science that elaborates the abstract structure that all progressions have in common merely in virtue of being progressions. It is not a science concerned with particular objects—the numbers.9

Let me give a brief flavour of his argument. The number ‘three’ may be represented by the symbols I I I or 3. One may construct the number using the supposedly more fundamental ideas of set theory in at least two different ways. All of these methods of expressing numbers yield the formula 4 + 1 = 5, sometimes as a theorem and sometimes as a definition of 5 or of +. Similarly with other rules of arithmetic. There seems to be no way of persuading a sceptic that any of these expressions for the number is more fundamental than any other. Benacerraf concludes that ‘three’ cannot be any of the expressions, and that one can use any progression of symbols or words to develop an idea of number. Let us nevertheless concede for the moment that small or ‘counting’ numbers exist in some sense, on the grounds that we can point to many different collections of (say) ten objects, and see that they have something in common. The idea that the Number System as a whole is a social construct seems to lead one into fundamental difficulties. I will examine these one at a time. If one is prepared to admit that 3 exists independently of human society then by adding 1 to it one must believe that 4 exists independently. Continuing 100 in this way seems to force the eventual conclusion that 1010 exists independently. But as a matter of fact it is not physically possible to continue repeating 100 the argument in the manner stated until one reaches the number 1010 . We must not be misled by the convention under which mathematicians pretend that this is possible ‘in principle’. If one does not believe that huge numbers exist independently then how can they have objective properties? The answer to this question is similar to that for chess. Constructed entities do indeed have properties, and while some of these may just be conventions, others may not be under our control. We decide on rules which we will obey in chess, and then we play according to the rules. Our agreement about the truth of theorems is of the same type as the agreement of people in the chess world about the correctness of a solution to a chess problem. One difference between mathematics and chess-players is that mathematicians are constantly altering the rules to see if we can find more interesting games. However, mathematicians may only change the rules within certain conventionally prescribed limits or they are deemed no longer to be doing mathematics. For example they cannot make the rules depend upon whether or not it is a religious holiday. Nor can they make the rules depend upon the geographical location of the practitioner, as can lawyers. Pure mathematicians reject issues relating to religion, race, nationality, gender, and even views about the structure of the world as valid bases for arguments, so it is not entirely surprising that they have been able to achieve a considerable consensus on the very rarified world remaining. Once one progresses sufficiently far in the creation of any social structure, whether it be

Arithmetic 83

mathematics or law, it takes on a life of its own, dictating what can and cannot be done. Every now and again controversies arise even in mathematics, but they may be examined for years or even decades before a consensus emerges. Even then the issues involved may be raised again if it appears worthwhile to do so; as time passes the task becomes ever greater because of the accumulated work based on the dominant tradition. There is a final question. If one does not believe that many of the entities in mathematics have an independent existence, how does one account for the extraordinary success of mathematics in explaining the world? There cannot be a simple answer to this question, to which we will return in the concluding section of the book. One part of the answer is that we understand the universe to the extent that we can predict its behaviour. Our ‘extraordinary success’ is only extraordinary by standards which we ourselves have set. We need to keep reminding ourselves that there exist chaotic phenomena which we will never be able to predict whether or not we use mathematical methods. Our own existence, both as a species and individually, depends upon historical contingencies whose details could not possibly be explained mathematically. While new scientific theories will certainly be developed, we do not expect these to be able to bypass the above problems. Quantum theory indicates that at a small enough scale prediction is fundamentally impossible, except in a probabilistic sense. We will see some of the evidence which justifies these claims in later chapters.

Notes and References [1] A similar division was described by Paul Bernays [Bernays 1998], in an article discussing the philosophical status of numbers in 1930–31. It is also possible to provide an empiricist defence for the existence of such a division. See Davies 2003 and Gillies 2000a. [2] Dunmore 1992, p. 218 [3] Acerbi 2000 [4] Bernays 1998, p. 253 [5] Davies 2003 [6] In technical terms I am suggesting a realist ontology for small numbers and an anti-realist ontology for large enough numbers. [7] See Acerbi 2000 for references to the literature on the ‘Sorites’. [8] Dummett 1978, p. 248–268 [9] Benacerraf 1983, p. 291. Benacerraf himself and several other philosophers criticized the argument of this paper subsequently. See Morton and Stich, 1996.

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4 How Hard can Problems Get?

HEALTH WARNING The next two chapters contain some genuine mathematics. If you are allergic to this, hold your breath and pass as quickly as possible through the affected areas.

Introduction The stock portrait of a pure mathematician is of a thin, introverted person, who is socially inept and likes to sit alone contemplating unfathomable mysteries. There is more than a grain of truth in this image. On the other hand I know mathematicians who are continuous balls of energy. A few have acquired the status of prophets in their own lifetimes, and are regularly surrounded by rings of disciples. Some have long term goals towards which they direct their energies for years on end. Yet others spend their lives hacking through a jungle, hoping to find something of interest if only they persist for long enough. The one thing which unites all these different people is incurable optimism. Not that this is obvious! Gödel proved that there are mathematical problems which are insoluble by normal methods of argument, but all mathematicians are sure that their own particular concern does not fall within this category. Indeed they have immense faith that if they persist long enough they will surely make some progress in resolving the issue to which they are devoting their energies. Roger Penrose based his popular books on the argument that while Gödel’s theorem constrains computing machines, human beings can transcend its limitations. Put briefly they can ‘see’ the truth without the need for chains of logical argument. To explain this he postulates that microtubules in neurons allow the influence of quantum effects on conscious thought. I do not have the expertise to judge whether microtubules and consciousness have some deep connection, and am happy to leave time to judge that issue. I am, however, less happy with Penrose’s belief that human beings have potentially unlimited powers of insight. Indeed it strikes me as astonishing, since all of our other bodily organs have obvious limits on their capacities. However, what different people do or

86 Introduction

do not find incredible is of less significance than what we discover when we look at the evidence. In this chapter I describe a few of the outstanding mathematical discoveries which have taken place during the last half century. They were not selected randomly: each of them says something about how far human mathematical powers extend. This, rather than their mathematical content, is also what I concentrate on when discussing them. Together they suggest that we are already quite close to our biological limits as far as the difficulty of proven theorems is concerned. This should not be taken as indicating that mathematics is coming to an end. New fields are constantly opening up, and these always start with ideas which are much more easily grasped than those of longer established fields. It is likely that interesting new mathematics will continue to appear for as long as anyone can imagine, because we will constantly discover new types of problem. This, however, is quite a different issue. When mathematicians talk about hard problems, we may mean one of several things. The first relates to problems which are hard in the mundane sense that great ability and effort are needed to find the solutions. The second sense is more technical and will be explained in the section on algorithms. There are finally statements which are undecidable within a particular formal system in the sense of Gödel; we will not discuss Gödel’s work extensively since much (possibly too much) has already been written about the subject. The remainder of this chapter may be omitted without serious loss. The topics which I have chosen are completely independent, and you are free to read any or all as you wish. (There is no examination ahead!) Before considering very hard problems let us look at one of intermediate difficulty. Pure mathematics and in particular arithmetic are often said to be a priori in the sense that the truths of theorems do not depend upon any empirical facts about the world. It is sometimes said that even God could not stop the identity 32 + 42 = 52 from being true! In 1966 Lander and Parkin discovered the identity 275 + 845 + 1105 + 1335 = 1445 by a computer search,1 thus disproving an old conjecture of Euler that the equation a 5 + b5 + c5 + d 5 = e5 has no solutions such that a, b, c, d, e are all positive whole numbers. The solubility of this equation is an example of an a priori fact. On the other hand it has a definite empirical tinge, in the sense that the solution was only discovered by a computer, and verifying that it is indeed a solution would involve about six pages of hand calculations. I know of no proof of solubility which provides the type of understanding a mathematician always seeks, and there is no obvious reason why a simpler proof should exist.

How Hard can Problems Get? 87

Fig. 4.1 The Welsh Local Authorities

The Four Colour Problem The four colour problem concerns the number of colours needed to cover a large plane area divided up into regions (a map) in such a way that no two neighbouring regions have the same colour. The conjecture is (or rather was) that four colours suffice for any conceivable map. The problem was formulated by Guthrie in 1852, and over the next hundred years a number of incorrect proofs of the conjecture were found. In 1976 Appel and Haken used a combination of clever mathematical ideas with lengthy computer calculations to provide a genuine proof. There were some blemishes in their first published solution, but these were later corrected. Their method could not involve an enumeration of all possible cases, since there are infinitely many maps. They devised an ingenious procedure to reduce the problem to one which could be solved in a finite length of time. Unfortunately they were not able to solve it by hand because the finite problem still involved too many cases, and they had to use 1200 hours of computer time to complete the proof. In spite of subsequent simplifications of the method,

88 Goldbach’s Conjecture

the original proof was never fully checked by other mathematicians. Recently an independent but related proof needing considerably less computer time has been completed by Robertson, Sanders, Seymour, and Thomas. It therefore seems almost certain that the theorem is true, but its proof is still not fully comprehensible. It is only fair to say that many mathematicians reacted rather negatively to this proof of the four colour theorem. In their view the issue was not whether the theorem was true, but why it was true (if indeed it was). The computer here acts as an oracle: it tells you the answer, but it is beyond your powers to check its calculations. If mathematics is about understanding, that is human understanding, then no satisfactory solution of the problem has yet been found. Tymoczko put it differently: the proof of the four colour theorem marks a fundamental philosophical shift in mathematics. It makes the truth of at least one theorem an empirical matter, in the sense that we have to rely on evidence from outside our own heads to complete the argument.2 What of the future? One possibility is that more and more problems will be discovered whose solution can only be obtained by an extensive computer-based search. Many mathematicians fervently hope that this will not happen, but it is entirely plausible.

Goldbach’s Conjecture This famous conjecture, proposed by Goldbach in a letter to Euler in 1742, states that every even number greater than 4 is the sum of two odd primes. Its truth is still unknown, although a large number of similar conjectures have now been proved. The conjecture has been confirmed for all numbers up to 1014 , which would be sufficient evidence of its truth for anyone except a mathematician. It has also been proved that it is asymptotically true in the sense that if one lists the exceptions (assuming that there are some), they become less and less frequent as one progresses.3 It may turn out that Goldbach’s conjecture is similar to the four colour theorem, and that a proof will depend upon a large computer search. Zeilberger even describes the possibility that mathematics might develop into a fully empirical science. I can envisage a paper of c.2100 that reads: We can show in a certain precise sense that the Goldbach conjecture is true with probability larger than 0.99999 and that its complete proof could be determined with a budget of $10 billion.4

There is, however, a worse possibility. Suppose that Goldbach’s conjecture is false and that 1324110300685 (a million further digits) 75692837093348572 is the smallest number which cannot be represented in such a way. Suppose also that the shortest way of proving the falsity is by a brute force search. In such a

How Hard can Problems Get? 89

case it is unlikely that the human race will ever discover that the conjecture is false, even if we are allowed to make full use of computers as powerful as we will ever have.

Fermat’s Last Theorem Fermat’s last theorem (FLT) is the proposition that it is impossible to find positive integers a, b, c and an integer m ≥ 3 such that a m + bm = cm . In 1637 Fermat wrote a marginal note in a book claiming that he had found a proof that his equation was insoluble. Nobody now takes his claim seriously, although there is no reason to doubt his sincerity. An enormous amount of work on the problem eventually led to the result that if Fermat’s equation does have a solution with m ≥ 3 then m ≥ 1000000. Many editors of mathematical journals received papers claiming to have found proofs of FLT. The one which sticks in my memory came from someone who claimed that the problem was mis-stated. Fermat was supposed to have claimed that there did not exist positive numbers a, b, c such that a m +bm = cm for all m ≥ 3. There are three problems with this theory. Firstly it is wrong. Secondly this new version is entirely trivial. Thirdly no mathematician cared what Fermat had written, or even whether he had ever existed. The point is rather that what is called FLT is a very interesting and deep problem whether or not it was devised by Fermat! Mathematicians use names as labels, and regularly attribute theorems to people who would not have understood even the statements, let alone the proofs. Fermat’s problem is not simply an isolated puzzle, of interest only to number theorists. It is part of a subject called the arithmetic of elliptic curves, which has ramifications throughout mathematics. Indeed elliptic curves provide the best current algorithms for factoring large integers, a matter of enormous practical importance in modern cryptography. In June 1993 Andrew Wiles, a British mathematician working in Princeton, New Jersey, came out of a period of about seven years of near monastic seclusion to give a lecture course on elliptic curves at the Isaac Newton Institute in Cambridge, England. At the end of this he announced that he had solved Fermat’s problem! The news appeared in newspapers all over the world, making him an instant celebrity, a unique achievement for a pure mathematician. Wiles acknowledged a serious error in the proof in December 1993, but with the help of Richard Taylor he patched this up within a year and the result was solid. This is one of the hardest mathematical problems solved up to the present date. The proof is beyond the intellectual grasp of most of the human race, and would take about ten years for a particularly gifted 18 year old to understand. This problem took over three hundred years from its initial formulation to its solution, during which period many partial results and insights were obtained.

90 Finite Simple Groups

When the time was ripe it needed about seven years out of the life of one of the most outstanding mathematicians of the twentieth century to obtain the solution. But at least the result could be grasped in its entirety by a single person of sufficient ability and dedication. Our next example is far worse in this respect.

Finite Simple Groups A group is a mathematical object containing a number of points (elements) in which multiplication and division are defined, but not addition. Groups are of major importance in mathematics for the description of symmetries, or rotations, of objects. There are 60 rotations of the dodecahedron below (Figure 4.2) which take it back to exactly the same position, including five around the axis shown. Other polyhedra, even those in higher space dimensions, have their own symmetry groups. Mathematicians have long wanted a complete list of symmetry groups. Among these are some which are regarded as the most fundamental, or ‘simple’, because they cannot be reduced in size in a certain technical sense. In 1972 David Gorenstein laid out a sixteen point programme for the classification of finite simple groups, and by the end of the decade a worldwide

Fig. 4.2 Rotation of a Dodecahedron

How Hard can Problems Get? 91

collaboration under his leadership had led to the solution of the problem. The final list can be written down in a few lines, and contains a small number of exceptional, or sporadic, groups, the biggest of which is called the Monster. This can be regarded as the rotation group of a polyhedron, but in 196883 dimensions rather than the usual three dimensions of physical space!

cyclic groups of prime order alternating groups on at least five letters groups of Lie type 26 exceptional groups.

Although the list is short, the complete proof was thousands of pages long, and some crucial aspects were never completed (a mathematician’s way of saying that there were serious mistakes in some of the papers). A new project to write out a simplified proof is likely to involve twelve volumes and more than 3000 pages of print. We have described a theorem whose proof only exists by the collective agreement of a community of scholars. In 1980 none of them understood the entire structure, and each had to trust that the others had done their respective parts thoroughly. Since then the amount and variety of confirming evidence makes it essentially certain that the basic results in the theory are correct, even if individual parts of the proof are faulty. Mathematics has certainly changed since the time of the classical Greeks!

A Practically Insoluble Problem What lessons can we learn from these examples? It is already the case that understanding the proofs of some theorems takes much of the working lives of the most mathematically able human beings. Extrapolating from these examples, there is no reason to believe that all theorems which are provable ‘in principle’ are actually within the grasp of sufficiently clever humans. We next give examples of (admittedly not very interesting) statements which are unlikely ever to be proved or disproved. The first uses the number π, most easily defined as equal to the circumference of a circle of diameter 1. The problem depends upon being able to compute the digits of π successively, but this is in principle straightforward, and the first hundred billion digits have indeed been computed. The first five hundred are

92 A Practically Insoluble Problem

given below. π ∼ 3.14159265358979323846264338327950288419716939937510 58209749445923078164062862089986280348253421170679 82148086513282306647093844609550582231725359408128 48111745028410270193852110555964462294895493038196 44288109756659334461284756482337867831652712019091 45648566923460348610454326648213393607260249141273 72458700660631558817488152092096282925409171536436 78925903600113305305488204665213841469519415116094 33057270365759591953092186117381932611793105118548 07446237996274956735188575272489122793818301194912 We will call a number n untypical if the nth digit of π and the 999 digits following that are all sevens.5 To find out whether a particular number is untypical one carries out a routine calculation which is bound to yield a positive or negative answer within a known length of time. In spite of the above, the simplest questions about such numbers cannot be answered at present, and may well never be answerable. It is not even known whether there are any untypical numbers. Here are arguments in favour of the two extreme possibilities. If one computes the first one hundred billion digits of π one finds that no number smaller than 1011 is untypical. Thus (nonmathematical) induction suggests that there are no untypical numbers. On the other hand, the digits of π satisfy every test for randomness which has been applied to them, and if the digits were indeed random then it could be proved that a sequence of a thousand sevens must occur somewhere in the sequence of digits. It is an interesting fact that many mathematicians prefer the second argument to the first, in spite of the fact that logically it is even more shaky. It uses non-mathematical induction in that it refers to a finite number of other tests of randomness which imply nothing about the question at hand. Secondly the digits of π are certainly not random: they can be computed by a completely determined procedure in which randomness plays no part. I cannot refrain from commenting that in my first draft of the above paragraphs I referred to the chain 0123456789 instead of the chain of a thousand sevens. Unfortunately I did not know that it had been proved by Kanada and Takahashi in 1997 that this chain does occur in π. The 0 in its first occurrence is the 17,387,594,880th digit of the decimal expansion of π . There has been great progress in methods of computing the digits of π over the last ten years. However, such developments cannot possibly enable us to decide whether the decimal expansion of π contains a sequence of a thousand consecutive 7s. To demonstrate this, let me make three assumptions. The first is that computational and theoretical progress may one day be so great that it becomes possible to determine a trillion coefficients of the decimal expansion of π every second, with no reduction in speed as one passes along the list of coefficients, however far one has to go. The second is that the only method of proving the existence

How Hard can Problems Get? 93

of the required sequence is by a brute force search. The third is that the first occurrence of the sequence is more or less where it would be for a completely random sequence of digits. Under these assumptions, none of which is proved of course, we can estimate how long it would take to find the first occurrence of the sequence. I will make no attempt at rigour, and the conclusion can only be regarded as an extremely rough approximation. If we break the expansion of π up into blocks of length 1000, then the chance that a particular block consists entirely of 7s is 1 in 101000 , so one would expect to have to consider something like 101000 blocks in order to find its first occurrence; of course the sequence may not occupy a single block neatly, but this problem may be taken into account. We need not compute every digit of π, but must compute at least one in every block of 1000, because if we leave any block unexamined we may have missed the sequence. Under our standing assumptions we then find that we probably have to compute of order 101000−3 digits and this will take us about 10985 seconds. This is vastly longer than the age of the universe, so we had better hope that one of the above assumptions is wrong (if we hope to find the sequence). A Platonic mathematician would say that either there exists an untypical number or there does not. This view is certainly psychologically comfortable, but it is not necessary to accept it in order to be a mathematician. Intuitionists would only say that such a number existed if they knew one or had a finite procedure which would definitely find one. They would only say that there was no such number if they could derive a contradiction from its existence. If neither was (currently) the case, they would remain silent. They would say that to do otherwise would be to adopt a purely philosophical position which would not increase human knowledge. We will discuss this in more detail in the next chapter. Warning To make a claim that a mathematical problem will never be solved is perhaps foolhardy. The eminent mathematician Littlewood once wrote that ‘the legend that every cipher is breakable is of course absurd, though widespread among people who should know better’. He proceeded to describe an ‘unbreakable’ code based upon a public coding procedure, a public book of log tables and a private key word of five digits. Fifty years later his code could be broken by standard desktop computer in a few minutes! I hope and believe that I am on safer ground than he was. If I am wrong either computation or mathematics will have advanced beyond the wildest dreams of current mathematicians.

Algorithms In this section we will discuss some problems which are hard in the sense of computational complexity. Some of these are completely soluble by carrying out a systematic search through all possibilities. However, this method of approach is often completely unrealistic not only for present-day computers but for any computers which could ever be designed. The examples all relate to

94 Algorithms

the behaviour of certain types of algorithm. To make sure that we start from a common position, let me describe an algorithm as a procedure which is applied repeatedly and systematically to an input of a given type. This description is rather forbidding and we start with a simple but famous example. The Collatz algorithm has as input a single number n. It carries out the following procedures: If n is even replace n by n/2. If n  = 1 is odd replace n by 3n + 1. If n = 1 then stop and print YES. If we start with n = 9 then the Collatz algorithm successively yields the values 9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1 so the sequence stops after 19 steps and prints YES. All algorithms of interest to us have a stopping condition, and when it is satisfied they print an output, which in this case can only be the word YES. For other algorithms there may be several possible different outputs. An algorithmic solution to a certain kind of problem is an algorithm which is guaranteed to provide the solution to all problems of the specified type. The Collatz problem is whether the Collatz algorithm stops after a finite number of steps, whatever value of n you start from. Surprisingly the answer to this problem is not known, although the algorithm does stop for all n up to 1012 . It might seem that one can settle this problem simply by running the algorithm and waiting, and indeed this is true for those values of n for which the algorithm does indeed stop. However, if there exists a value of n for which the sequence is infinite, then this cannot be discovered by use of the algorithm. The fact that it has not stopped after 1012 steps says nothing about what might happen after more steps. No solution to the Collatz problem is known, and it is not likely that the situation will change soon. There are problems which are algorithmically undecidable: there is no systematic way of solving all the problems of the specified type. This is a very strong statement, much stronger than saying that no algorithm has yet been discovered. It only makes sense if one is absolutely precise about what counts as an algorithm, but this has been done in a way which commands general assent. It can then be proved absolutely rigorously that algorithmically undecidable problems exist. We will not discuss this issue further, since it is very technical and has been treated in great detail in several other places. Let us return to the simplest type of algorithmic problem, one for which it is quite clear that it can be solved in a finite length of time just by testing each one of a large but finite number of potential solutions. Algorithms for such problems

How Hard can Problems Get? 95

are divided into two types, which we will call Fast and Slow.6 All useful algorithms are Fast, but some Fast algorithms are not fast enough to be useful. The speed of an algorithm depends upon how one decides to measure the size of the input and also how one defines steps/operations. For most purposes one regards multiplications and additions of large numbers as single operations, and just asks how many are performed. However the amount of time spent transporting data to and from the memory of the computer is also important and one might include such acts as operations. If we ask for the number of multiplications needed to compute 2n , that is 2 times itself n times, it seems obvious that the answer is n. However, there is a much better method which involves radically fewer multiplications. Namely we write 2×2=4 4 × 4 = 16 16 × 16 = 256 256 × 256 = 65536 65536 × 65536 = 4294967296 which yields 232 in just 5 multiplications. One can actually compute 2n for general n using far fewer than n multiplications.7 Given that a problem may be solved in various different ways, it is obviously desirable to find the most efficient possible way. A problem is said to be (computationally) hard if the number of operations needed to solve it increases extremely rapidly as the size of the problem increases, for all possible algorithms. This is clearly difficult to know. It may be that every algorithm currently known for solving a particular problem is Slow, and that nobody believes that a faster algorithm can be found, but that is different from proving that no Fast algorithm can ever be found. There is one respect in which the idea of thinking of a multiplication as a single operation is misleading. Suppose we have two numbers, one with m digits and the other with n digits. If m and n are large enough then a normal processor cannot multiply them in one step, and they have to be treated as long strings of digits. One possibility is to multiply them using a computer analogue of primary school long multiplication. The number of multiplications and additions of digits is of order m × n. So every algorithm involving sufficiently large numbers is actually much slower than our previous discussion indicated. The above analysis of algorithms suggests that the only issue in the design of algorithms is to minimize the number of elementary arithmetic operations. This is far from being the case. In all early computers and many current computers there is only one processor, so arithmetic operations do indeed have to be carried out one at a time. However, parallel computers have many processors, so one can carry out multiple operations simultaneously provided one can find an appropriate way of organizing the computation and managing the flow of information between processors.

96 How to Handle Hard Problems

Suppose one wants to add 512 different numbers (e.g. salaries). The obvious way of doing this takes 511 clock cycles (computers run on a very precise schedule of one operation per clock cycle). If, however, one has an unlimited number of processors, one can add the numbers in pairs in the first clock cycle leaving only 256 numbers to add. In the second clock cycle one can then add the remaining 256 in pairs. Continuing this way the task is finished in 9 clock cycles. There are many problems in implementing this idea. Firstly most of the processors spend most of their time doing nothing, which is very wasteful. Secondly all of the data has to be moved to the appropriate processors before the computation can start, whereas in the normal algorithm one only needs to bring one item at a time. Thirdly it might not be possible to parallelize some problems at all. Nevertheless the size of many computations in physics is now so large that enormous efforts are being made to find ways of solving the communication and other design problems associated with building large parallel machines. From a purely theoretical point of view the difficulty of an algorithm is now seen to depend on the computer architecture as much as on the problem itself.

How to Handle Hard Problems Sometimes a problem is extremely hard to solve in the sense that the only known algorithms for solving it are very slow. Two methods for sidestepping this problem have been devised. The first is that one may ask not for the best solution but merely for a good enough solution. Here is an example. Define n! to be the result of multiplying all the integers 1, 2, 3, 4, . . . , (n−1), n together. To evaluate this we need to perform n multiplications. On the other hand Stirling’s formula provides an extremely good approximation to n! which may be computed far more rapidly.8 It enables one to obtain 1000! ∼ 4.0238726 × 102567 with only 10 operations if one regards taking a power as a single operation, or about 30 operations if we use the method already described for computing powers. The following table shows that Stirling’s formula is extraordinarily good even for very small numbers: n n! Stirling

1 1 1.002

2 2 2.001

3 6 6.001

4 24 24.001

5 120 120.003

6 720 720.009

This illustrates the general fact that if one is prepared to compromise a little on the accuracy or quality of a solution, a problem may become radically easier. The second method of evading intractable problems is probabilistic. The most famous case of this is finding whether a very large (e.g. hundred-digit) number is a prime. One cannot simply divide the number by all smaller numbers

How Hard can Problems Get? 97

in turn and see if the remainder ever vanishes. The task would take the lifetime of the Universe even for a single thousand-digit prime. In 1980 Michael Rabin devised a probabilistic procedure which solves this problem rapidly but with an extremely small chance of giving the wrong answer. This is now used in commercial encryption systems which transfer money between banks and over the internet. I will not (indeed could not!) describe the procedure, but refer to page 178, where a different and much simpler probabilistic algorithm is described. It marks another step in the transformation of mathematics into an empirical science. In the last few weeks a Fast (deterministic polynomial) procedure for deciding whether a given number is a prime, without using probability ideas, has been announced by Agrawal et al.9 The simplicity of this algorithm came as a shock to the community, but it appears to be correct. Such discoveries, and the possible new vistas they open up, are among the things which make it such a joy to be a mathematician. Fortunately (or unfortunately depending on your political views) it does not affect the security of the RSA encryption algorithm. Nor, in practice, is it faster than the existing probabilistic algorithms, but who can tell what future developments might bring?

Notes and References [1] Hollingdale 1989, p. 148 [2] Tymoczko 1998 [3] Technically the statement is that if m(n) is the number of exceptions less than n then limn→∞ m(n)/n = 0. [4] Zeilberger 1993 [5] I have taken the idea for this example from Gale 1989, but it goes back to Brouwer in the 1920s. [6] We say that an algorithm is Fast, or polynomial, if for every problem of size n the algorithm solves the problem in at most cnk steps, for some constants c, k which do not depend on n. Both c and k may be of importance for problems of medium size but for very large problems the value of k is usually more significant. [7] The actual number of multiplications needed is the smallest number greater than or equal to log2 (n). [8] This formula, namely 1

n! ∼ (2π)1/2 nn+1/2 e−n+ 12n is usually attributed to the eighteenth century Scottish mathematician James Stirling. It was actually discovered by de Moivre, who used it for applications in probability theory. [9] Agrawal et al. 2002

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5 Pure Mathematics

5.1

Introduction

The goal of this chapter is to demolish the myth that mathematics is uniquely free of controversy, and therefore a guaranteed source of objective and eternal knowledge. To be sure, attitudes in the subject generally change very slowly. At any moment there seems to be an overwhelming consensus, provided one excludes a few mavericks. However, this consensus has changed several times over the last two hundred years. Among the debates within the subject one of the most important concerned its foundations. This was most active in the period between 1900 and 1940. It led to an enormous amount of interesting work in logic and set theory, but not to the intended goal. Indeed the foundations were seen to be in a more unsatisfactory state at the end of the period than they had appeared to be at the start. We describe how this came to pass. I believe that we are now in the early stages of yet another, computer-based, revolution. Some of my colleagues may disagree, but when one of the lines of investigation into the Riemann hypothesis in number theory involves examining the statistics of millions of numerically computed zeros, something has surely changed. We will discuss this further near the end of the chapter. Mathematicians themselves rarely have any regard for the historical context of their subject. They attach names to theorems as mere labels, without any interest in whether the people named could even have understood the statements of ‘their’ theorems. Each generation of students is provided with a more streamlined version of the subject, in which the concepts are presented as if no other route was possible. The order in which topics are presented in a lecture course may jump backwards and forwards hundreds of years when compared with the order in which they were discovered, but this is almost never mentioned. Of course this is defensible: mathematics is a different subject from the history of mathematics. But the result is to leave most mathematics students ignorant of the process by which new mathematics is created. I hope that what follows will help a little to correct this imbalance.

100 Origins

5.2

Origins

The origins of mathematics are shrouded in mystery. One of our earliest sources of information comes from the discovery of hundreds of thousands of clay tablets bearing cuneiform text in Mesopotamia. A few hundred contain material of mathematical interest. From them we glean many interesting but isolated facts about the knowledge of the Babylonians as early as 2000 bc. Among these are their creation of tables of squares and cubes of the numbers up to 30 and their ability to solve quadratic equations. They explained their general procedures using particular numerical examples, since they had no algebraic notation in the modern sense. One of the tablets, dating from about 1600 bc, contains the extremely good approximation 1+

24 60

+

51 602

+

10 603

∼ 1.4142130

(in our notation) to the square root of 2. The tablet called Plimpton 322, dating from before 1600 bc, shows that they had a method for generating Pythagorean triples such as 32 + 42 = 52 and 1192 + 1202 = 1692 long before the time of Pythagoras. Such triples were familiar in China and India at a very early date, and there is some evidence for a common origin of this and other mathematical knowledge. For many people mathematics means formulating general propositions and proving them by logical arguments from some agreed starting point. In this sense mathematics started in classical Greece, as did so many other aspects of our civilization. After that glorious but brief period centuries were to pass before the subject changed substantially. From about 800 ad Arabic mathematicians started a major development of algebra, arithmetic, trigonometry, and many other areas of mathematics. These percolated slowly through to Europe, and were often described as European inventions until quite recently. During the seventeenth century the focus of development shifted decisively to Europe, where it stayed until some time in the twentieth century. This section describes the historical development of geometry, and the changing philosophical beliefs about its status over the last two hundred years. Later sections discuss logic, set theory and the real number system from a similar point of view.

Greek Mathematics The main codification of the Greeks’ work in geometry was due to Euclid around 300 bc, but Archimedes’ importance as an original thinker was certainly much greater. Euclid’s Elements appeared in thirteen books, with two later additions

Pure Mathematics 101

by other authors. These were preserved during the European dark ages by the Arabs; the first translation available in Europe was that of Adelard in 1120. The achievement of the Greeks in geometry was revolutionary. They transformed the subject into the first fully rigorous mathematical discipline based upon precisely stated assumptions (called axioms), and proceeded to build a massive intellectual structure using rigorous logical arguments. The method of proof which Euclid used was regarded as the model for all subsequent mathematics for almost two thousand years. Indeed Euclid was still taught in some schools in England in the mid-twentieth century. We have already encountered the mysterious number π . In the ancient world this was often approximated by 3 or 22/7. Among Archimedes’ claims to fame is the first serious attempt to evaluate it accurately. By putting a regular polygon with 96 sides inside the circle, and a similar one outside, he was able to prove rigorously that 223 71